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<title>Rotationskörper</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Rotationskörper</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p><b>Rotationskörper</b> wird in der <a href="Geometrie" title="Geometrie">Geometrie</a> ein <a href="K%C3%B6rper_(Geometrie)" title="Körper (Geometrie)">Körper</a> genannt, dessen Oberfläche durch <a href="Rotationsbewegung" class="mw-redirect" title="Rotationsbewegung">Rotation</a> einer erzeugenden Kurve um eine <a href="Rotationsachse" title="Rotationsachse">Rotationsachse</a> gebildet wird (siehe <a href="Rotationsfl%C3%A4che" title="Rotationsfläche">Rotationsfläche</a>). Die Rotationsachse wird auch <i>Figurenachse</i> genannt.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Die Kurve liegt dabei in einer <a href="Ebene_(Mathematik)" title="Ebene (Mathematik)">Ebene</a>, und auch die Achse liegt in ebenderselben. Ein bekannter Rotationskörper ist der <a href="Torus" title="Torus">Torus</a>. Er wird durch die Rotation eines <a href="Kreis_(Geometrie)" class="mw-redirect" title="Kreis (Geometrie)">Kreises</a> gebildet. Auch <a href="Kegel_(Geometrie)" title="Kegel (Geometrie)">Kegel</a> und <a href="Zylinder_(Geometrie)" title="Zylinder (Geometrie)">Zylinder</a> sind Rotationskörper.

</p><p>Das <a href="Volumen" title="Volumen">Volumen</a> und die <a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Oberfläche</a> werden mit den sogenannten <b>Guldinschen Regeln</b><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> (benannt nach dem Mathematiker und Astronomen <a href="Paul_Guldin" title="Paul Guldin">Paul Guldin</a>) errechnet. Bereits in der <a href="Antike" title="Antike">Antike</a> waren diese als <b>Baryzentrische Regeln</b> oder <b>Zentrobarische Regel</b> bekannt und wurden vom griechischen Mathematiker <a href="Pappos_von_Alexandria" class="mw-redirect" title="Pappos von Alexandria">Pappos von Alexandria</a> beschrieben.

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<div class="mw-heading mw-heading2"><h2 id="Berechnung_des_Volumens_eines_Rotationskörpers"><span id="Berechnung_des_Volumens_eines_Rotationsk.C3.B6rpers"></span>Berechnung des Volumens eines Rotationskörpers</h2></div>

<p>Falls die erzeugende Kurve die Drehachse schneidet, ist zu überlegen, ob die entsprechenden Teilvolumina als positive oder negative Beiträge zum Gesamtvolumen gezählt werden sollen.

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<div class="mw-heading mw-heading3"><h3 id="Rotation_um_die_x-Achse">Rotation um die <i>x</i>-Achse</h3></div>

<p>Für einen Rotationskörper, der durch Rotation der Fläche, die durch den Graphen der Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> im Intervall <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"></span>, die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-Achse und die beiden Geraden <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

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<annotation encoding="application/x-tex">{\displaystyle x=a}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=b}">

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<annotation encoding="application/x-tex">{\displaystyle x=b}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229dd3f0f42d375fa257bdc1389f92f7b225c415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\displaystyle x=b}" loading="lazy"></span> begrenzt wird, um die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-Achse entsteht, lautet die Formel zur Volumenberechnung:

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<annotation encoding="application/x-tex">{\displaystyle V=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/999453555f00196dcf803c9de0de892e7ecb5dd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.202ex; height:6.343ex;" alt="{\displaystyle V=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Rotation_um_die_y-Achse">Rotation um die <i>y</i>-Achse</h3></div>

<div class="mw-heading mw-heading4"><h4 id="1._Fall:_„disc_integration“"><span id="1._Fall:_.E2.80.9Edisc_integration.E2.80.9C"></span>1. Fall: „disc integration“</h4></div>

<p>Bei Rotation (um die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>-Achse) der Fläche, die durch den Graphen der Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> im Intervall <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

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<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"></span>, die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>-Achse und die beiden Geraden <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(a)}">

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<annotation encoding="application/x-tex">{\displaystyle y=f(a)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee21b9f4ce2c47212428e4db192cc36cac0ea45c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.572ex; height:2.843ex;" alt="{\displaystyle y=f(a)}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(b)}">

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<annotation encoding="application/x-tex">{\displaystyle y=f(b)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c7e02035b9f4bd804159a2779366692dc769f4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.339ex; height:2.843ex;" alt="{\displaystyle y=f(b)}" loading="lazy"></span> begrenzt wird, muss man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=f(x)}">

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<annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2311a6a75c54b0ea085a381ba472c31d59321514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" loading="lazy"></span> umformen zur <a href="Umkehrfunktion" title="Umkehrfunktion">Umkehrfunktion</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=f^{-1}(y)}">

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<annotation encoding="application/x-tex">{\displaystyle x=f^{-1}(y)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8c259d877499b841cadd9d19146f2413cec1882.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.046ex; height:3.176ex;" alt="{\displaystyle x=f^{-1}(y)}" loading="lazy"></span>. Diese existiert, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> <a href="Stetige_Funktion" title="Stetige Funktion">stetig</a> und <a href="Streng_monotone_Funktion" class="mw-redirect" title="Streng monotone Funktion">streng monoton</a> ist. Falls nicht (wie z. B. im Bild rechts oben), lässt sich <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> vielleicht in Abschnitte zerlegen, in denen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> jeweils stetig und streng monoton ist. Die zu diesen Abschnitten gehörenden Volumina müssen dann separat berechnet und addiert werden.

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<annotation encoding="application/x-tex">{\displaystyle V=\pi \cdot \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}(f^{-1}(y))^{2}\mathrm {d} y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b1ce64a62d48eec48397d1f0f8042a301309a961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.26ex; height:6.676ex;" alt="{\displaystyle V=\pi \cdot \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}(f^{-1}(y))^{2}\mathrm {d} y}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle y=f(x)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2311a6a75c54b0ea085a381ba472c31d59321514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.672ex; height:2.843ex;" alt="{\displaystyle y=f(x)}" loading="lazy"></span> substituiert, erhält man mithilfe der Substitutionsregel für Integrale für das Volumen um die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>-Achse

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<annotation encoding="application/x-tex">{\displaystyle V=\pi \cdot \int _{\min(a,b)}^{\max(a,b)}x^{2}\mathrm {d} y=\pi \cdot \int _{\min(a,b)}^{\max(a,b)}x^{2}\cdot \left|f'(x)\right|\mathrm {d} x}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3ae5eb04cba07cfdbf8d05aa98c0dd753d10ea8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:51.257ex; height:6.676ex;" alt="{\displaystyle V=\pi \cdot \int _{\min(a,b)}^{\max(a,b)}x^{2}\mathrm {d} y=\pi \cdot \int _{\min(a,b)}^{\max(a,b)}x^{2}\cdot \left|f'(x)\right|\mathrm {d} x}" loading="lazy"></span>.</dd></dl>

<p>Der Absolutwert von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f'}">

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<annotation encoding="application/x-tex">{\displaystyle f'}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/258eaada38956fb69b8cb1a2eef46bcb97d3126b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.005ex; height:2.843ex;" alt="{\displaystyle f'}" loading="lazy"></span> und die min/max-Funktionen in den Integralgrenzen sichern ein positives Integral.

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<div class="mw-heading mw-heading4"><h4 id="2._Fall:_„shell_integration“_(Zylindermethode)"><span id="2._Fall:_.E2.80.9Eshell_integration.E2.80.9C_.28Zylindermethode.29"></span>2. Fall: „shell integration“ (Zylindermethode)</h4></div>

<p>Bei Rotation (um die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>-Achse) der Fläche, die durch den Graphen der Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> im Intervall <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"></span>, die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-Achse und die beiden Geraden <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=b}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229dd3f0f42d375fa257bdc1389f92f7b225c415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\displaystyle x=b}" loading="lazy"></span> begrenzt wird, gilt die Formel:

</p>

</mrow>

</mrow>

</msubsup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=2\pi \,\int _{a}^{b}x\,f(x)\,\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2bd747eb6c1771a5f002b6987bfc551d13a28c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.087ex; height:6.343ex;" alt="{\displaystyle V=2\pi \,\int _{a}^{b}x\,f(x)\,\mathrm {d} x}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Guldinsche_Regeln">Guldinsche Regeln</h2></div>

<p>Die beiden guldinschen Regeln, benannt nach dem Schweizer Mathematiker <a href="Paul_Guldin" title="Paul Guldin">Paul Guldin</a>, verkürzen Oberflächen- und Volumenberechnungen von Rotationskörpern enorm, falls sich die <a href="Geometrischer_Schwerpunkt" title="Geometrischer Schwerpunkt">Linien- oder Flächenschwerpunkte</a> der rotierenden Objekte unter Ausnutzen der Symmetrien der jeweiligen Aufgabe einfach erkennen lassen (s. u. Torus-Beispiele).

</p><p>Bezeichnungen:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> = Oberfläche</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> = Rauminhalt</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> = Länge der erzeugenden Linie (Profillinie)</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> = Flächeninhalt der erzeugenden Fläche</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> = Radius des Schwerpunktkreises</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> = Radius des rotierenden Kreises (Torus-Beispiele)</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Erste_Regel">Erste Regel</h3></div>

<p>Der <a href="Fl%C3%A4cheninhalt" title="Flächeninhalt">Flächeninhalt</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> der <a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelfläche</a> eines Rotationskörpers, dessen Rotationsachse die erzeugende Linie nicht schneidet, ist gleich dem Produkt aus der Länge der erzeugenden Linie (Profillinie) und dem Umfang des Kreises (Schwerpunktkreis), der durch die Rotation des Linienschwerpunktes der Profillinie erzeugt wird:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=L\cdot 2\pi R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bed22cd849f559833673807bb4cfcdfd47560327.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.061ex; height:2.176ex;" alt="{\displaystyle M=L\cdot 2\pi R}" loading="lazy"></span></dd></dl>

<p>Ausgedrückt in Abhängigkeit von der Funktion <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> der erzeugenden Linie ergibt sich der Flächeninhalt als:

</p>

<div class="mw-heading mw-heading4"><h4 id="Bei_Rotation_um_die_x-Achse">Bei Rotation um die <i>x</i>-Achse</h4></div>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

<mrow>

<mrow>

<msup>

</msup>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=2\pi \int _{a}^{b}f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6c1b501cd69e7ea1cb968b3de87420f84d48207.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.07ex; height:6.343ex;" alt="{\displaystyle M=2\pi \int _{a}^{b}f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x}" loading="lazy"></span></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle R=y_{s}={\frac {1}{L}}\int _{L}y\mathrm {d} L}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b9239bf7e7f081db01a085bafc905aaed6b6303.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.313ex; height:3.509ex;" alt="{\displaystyle \textstyle R=y_{s}={\frac {1}{L}}\int _{L}y\mathrm {d} L}" loading="lazy"></span> als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy"></span>-Koordinate des Linienschwerpunktes der Linie <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> und ihrem Linienelement <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} L}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} L}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c41767d965adf9cd0a04cc7cf4b39e9d33fbaeb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.875ex; height:2.176ex;" alt="{\displaystyle \mathrm {d} L}" loading="lazy"></span> findet man

</p>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=L\cdot 2\pi R=L\cdot 2\pi \cdot {\frac {1}{L}}\int _{L}f(x)\mathrm {d} L=2\pi \int _{L}f(x)\mathrm {d} L}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a967a74c64502b6fa0a303fc76a0f984b7ece8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:53.029ex; height:5.676ex;" alt="{\displaystyle M=L\cdot 2\pi R=L\cdot 2\pi \cdot {\frac {1}{L}}\int _{L}f(x)\mathrm {d} L=2\pi \int _{L}f(x)\mathrm {d} L}" loading="lazy"></span>,</dd></dl>

<p>was das obige Ergebnis darstellt, wenn noch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \mathrm {d} L={\sqrt {(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}}={\sqrt {1+\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}}}\mathrm {d} x}">

</mrow>

<msqrt>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<msqrt>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \mathrm {d} L={\sqrt {(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}}={\sqrt {1+\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}}}\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/290c55d79935c7aad3c40eb17bbeed37b2b8feed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:40.503ex; height:6.176ex;" alt="{\displaystyle \textstyle \mathrm {d} L={\sqrt {(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}}={\sqrt {1+\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}}}\mathrm {d} x}" loading="lazy"></span> mit den <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-Intervallgrenzen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"></span> eingesetzt wird.

</p>

<div class="mw-heading mw-heading4"><h4 id="Bei_Rotation_um_die_y-Achse">Bei Rotation um die <i>y</i>-Achse</h4></div>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msqrt>

<msup>

<mrow>

<msup>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=2\pi \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}f^{-1}(y){\sqrt {1+\left[\left(f^{-1}(y)\right)'\right]^{2}}}\mathrm {d} y}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8743a689c8736b4e051c1f8dd100506be0460a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:50.318ex; height:6.676ex;" alt="{\displaystyle M=2\pi \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}f^{-1}(y){\sqrt {1+\left[\left(f^{-1}(y)\right)'\right]^{2}}}\mathrm {d} y}" loading="lazy"></span></dd></dl>

<p>Wie oben bei der Volumenberechnung muss auch hier gegebenenfalls die Rechnung für die stetigen und streng monotonen Abschnitte von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span>, in denen die Umkehrfunktion existiert, separat durchgeführt werden.

</p><p><b>Beispiel:</b> Oberfläche eines <a href="Rotationstorus" class="mw-redirect" title="Rotationstorus">Rotationstorus</a>:

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=2\pi r\cdot 2\pi R=4\pi ^{2}rR}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cf61f43554972fb3802221d4d5a37f8bc726831e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.483ex; height:2.676ex;" alt="{\displaystyle M=2\pi r\cdot 2\pi R=4\pi ^{2}rR}" loading="lazy"></span></dd></dl>

<p><i>Siehe auch:</i> <a href="Mantelfl%C3%A4che" title="Mantelfläche">Mantelfläche</a>

</p>

<div class="mw-heading mw-heading3"><h3 id="Zweite_Regel">Zweite Regel</h3></div>

<p>Das Volumen eines Rotationskörpers ist gleich dem Produkt aus dem Flächeninhalt der erzeugenden Fläche und dem Umfang des Kreises, der durch die Rotation des Flächenschwerpunktes dieser Fläche erzeugt wird:

</p>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=A\cdot 2\pi R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70532be2840e70032a97c499865a7b7c97625e42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.566ex; height:2.176ex;" alt="{\displaystyle V=A\cdot 2\pi R}" loading="lazy"></span></dd></dl>

<p>Im Folgenden wird die Rotation einer Fläche um die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-Achse betrachtet, der Fall einer gekippten Rotationsachse lässt sich durch <a href="Koordinatentransformation" title="Koordinatentransformation">Koordinatentransformation</a> erreichen. Im Fall der Rotation um die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-Achse einer Fläche zwischen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span>, der <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-Achse und den Grenzen <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=a}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aaae23950e96a955ab5b07015a168fd931d4d82b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.658ex; height:1.676ex;" alt="{\displaystyle x=a}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x=b}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=b}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/229dd3f0f42d375fa257bdc1389f92f7b225c415.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.426ex; height:2.176ex;" alt="{\displaystyle x=b}" loading="lazy"></span> ergibt sich das Volumen ausgedrückt durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> als Flächenschwerpunkt zu

</p>

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=A\cdot 2\pi {\tfrac {1}{A}}\int _{A}y\mathrm {d} A=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d73e0157a18747100b55056b36ebd234efb433cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.008ex; height:6.343ex;" alt="{\displaystyle V=A\cdot 2\pi {\tfrac {1}{A}}\int _{A}y\mathrm {d} A=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}" loading="lazy"></span></dd></dl>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y={\tfrac {f(x)}{2}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1da39cf92691f556c01724faceedc76ed8e7ad56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:8.214ex; height:4.176ex;" alt="{\displaystyle y={\tfrac {f(x)}{2}}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {d} A=f(x)\mathrm {d} x}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} A=f(x)\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec01e7c40f484b0b2b3cbb85b16b6fcc2abf652e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.174ex; height:2.843ex;" alt="{\displaystyle \mathrm {d} A=f(x)\mathrm {d} x}" loading="lazy"></span>.

</p><p><b>Beispiel:</b> Volumen eines Rotationstorus:

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\pi r^{2}\cdot 2\pi R=2\pi ^{2}r^{2}R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99759d6f600be229005133b66cbb9a99dcc31e1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:24.774ex; height:2.676ex;" alt="{\displaystyle V=\pi r^{2}\cdot 2\pi R=2\pi ^{2}r^{2}R}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Parameterform">Parameterform</h2></div>

<p>Wenn eine <a href="Kurve_(Mathematik)" title="Kurve (Mathematik)">Kurve</a> durch ihre Parameterform <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x(t),y(t))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x(t),y(t))}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b7b80868600db89fef84e4d41317b7c8a1e0d047.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.626ex; height:2.843ex;" alt="{\displaystyle (x(t),y(t))}" loading="lazy"></span> in einem <a href="Intervall_(Mathematik)" title="Intervall (Mathematik)">Intervall</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [a,b]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [a,b]}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c4b788fc5c637e26ee98b45f89a5c08c85f7935.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.555ex; height:2.843ex;" alt="{\displaystyle [a,b]}" loading="lazy"></span> definiert wird, sind die <a href="Volumen" title="Volumen">Volumina</a> der <a href="K%C3%B6rper_(Geometrie)" title="Körper (Geometrie)">Körper</a>, die durch Drehen der Kurve um die x-Achse oder die y-Achse erzeugt werden, gegeben durch<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{x}=\int _{a}^{b}\pi y^{2}\,{\frac {\mathrm {d} x}{\mathrm {d} t}}\,\mathrm {d} t}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a4e27ecdeff79907cccb0946a4c82c8280caab9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.326ex; height:6.343ex;" alt="{\displaystyle V_{x}=\int _{a}^{b}\pi y^{2}\,{\frac {\mathrm {d} x}{\mathrm {d} t}}\,\mathrm {d} t}" loading="lazy"></span></dd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V_{y}=\int _{a}^{b}\pi x^{2}\,{\frac {\mathrm {d} y}{\mathrm {d} t}}\,\mathrm {d} t}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4928866fda98bb6c2be3b0ec3309a3e02b739532.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.198ex; height:6.343ex;" alt="{\displaystyle V_{y}=\int _{a}^{b}\pi x^{2}\,{\frac {\mathrm {d} y}{\mathrm {d} t}}\,\mathrm {d} t}" loading="lazy"></span></dd></dl>

<p>Der <a href="Oberfl%C3%A4cheninhalt" class="mw-redirect" title="Oberflächeninhalt">Oberflächeninhalt</a> dieser <a href="K%C3%B6rper_(Geometrie)" title="Körper (Geometrie)">Körper</a> ist gegeben durch<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{x}=\int _{a}^{b}2\pi y\,{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}}}\,\mathrm {d} t}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60346d0e45c2c232b0abdecbbf81c33f18e163a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.728ex; height:7.676ex;" alt="{\displaystyle M_{x}=\int _{a}^{b}2\pi y\,{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}}}\,\mathrm {d} t}" loading="lazy"></span></dd>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msqrt>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

</msqrt>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{y}=\int _{a}^{b}2\pi x\,{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}}}\,\mathrm {d} t}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ced57135e3a8ee739d97ff7326923065e4f36d54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.779ex; height:7.676ex;" alt="{\displaystyle M_{y}=\int _{a}^{b}2\pi x\,{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}}}\,\mathrm {d} t}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Keplersche_Fassregel">Keplersche Fassregel</h2></div>

<p>Die <a href="Keplersche_Fassregel" class="mw-redirect" title="Keplersche Fassregel">Keplersche Fassregel</a> gibt

</p>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V={\frac {h}{6}}\cdot \left(q(0)+4q\left({\frac {h}{2}}\right)+q(h)\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/44cfffe58fa71832fd0b2a71639d7c2dde53c782.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.316ex; height:6.176ex;" alt="{\displaystyle V={\frac {h}{6}}\cdot \left(q(0)+4q\left({\frac {h}{2}}\right)+q(h)\right)}" loading="lazy"></span></dd></dl>

<p>als <a href="N%C3%A4herungswert" title="Näherungswert">Näherungswert</a> für das <a href="Volumen" title="Volumen">Volumen</a> eines <a href="K%C3%B6rper_(Geometrie)" title="Körper (Geometrie)">Körpers</a>, dessen <a href="Querschnittsfl%C3%A4che" class="mw-redirect" title="Querschnittsfläche">Querschnittsfläche</a> an drei Stellen bekannt ist, an. Ist der Körper ein Rotationskörper, so gilt bei Rotation um die <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-Achse:

</p>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/999453555f00196dcf803c9de0de892e7ecb5dd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.202ex; height:6.343ex;" alt="{\displaystyle V=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}" loading="lazy"></span></dd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

<msup>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \approx \pi {\frac {b-a}{6}}\cdot \left((r(a))^{2}+4\left(r\left({\frac {a+b}{2}}\right)\right)^{2}+(r(b))^{2}\right)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7824a2034df1b17ae03aab73cb0ed68842072483.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:50.799ex; height:7.509ex;" alt="{\displaystyle \approx \pi {\frac {b-a}{6}}\cdot \left((r(a))^{2}+4\left(r\left({\frac {a+b}{2}}\right)\right)^{2}+(r(b))^{2}\right)}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Rotationsfl%C3%A4che" title="Rotationsfläche">Rotationsfläche</a></li>

<li><a href="Kugel" title="Kugel">Kugel</a></li>

<li><a href="Kegel_(Geometrie)" title="Kegel (Geometrie)">Kegel</a></li>

<li><a href="Kegelstumpf" title="Kegelstumpf">Kegelstumpf</a></li>

<li><a href="Zylinder_(Geometrie)" title="Zylinder (Geometrie)">Zylinder</a></li>

<li><a href="Rotationsparaboloid" class="mw-redirect" title="Rotationsparaboloid">Rotationsparaboloid</a></li>

<li><a href="Rotationshyperboloid" class="mw-redirect" title="Rotationshyperboloid">Rotationshyperboloid</a></li>

<li><a href="Rotationsellipsoid" title="Rotationsellipsoid">Rotationsellipsoid</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Kurt Magnus: <cite style="font-style:italic">Kreisel</cite>. Theorie und Anwendungen. Springer, Berlin, Heidelberg 1971, ISBN 978-3-642-52163-8, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>44</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Rotationsk%C3%B6rper&rft.au=Kurt+Magnus&rft.btitle=Kreisel&rft.date=1971&rft.genre=book&rft.isbn=9783642521638&rft.pages=44&rft.place=Berlin%2C+Heidelberg&rft.pub=Springer" style="display:none"> </span></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text">Ilja N. Bronstein, Konstantin A. Semendjaew: <cite style="font-style:italic">Taschenbuch der Mathematik</cite>. 20. Auflage. Teubner; Nauka, Leipzig; Moskau 1981, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>369<span style="display:inline-block;width:.2em"> </span>f</span>. (XII, 860).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Rotationsk%C3%B6rper&rft.au=Ilja+N.+Bronstein%2C+Konstantin+A.+Semendjaew&rft.btitle=Taschenbuch+der+Mathematik&rft.date=1981&rft.edition=20.+Auflage&rft.genre=book&rft.pages=369+f.&rft.place=Leipzig%3B+Moskau&rft.pub=Teubner%3B+Nauka" style="display:none"> </span></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><span class="book">A. K. Sharma: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Application Of Integral Calculus</cite>. Discovery Publishing House, 2005, ISBN 81-7141-967-4, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>168</span> (englisch, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V_WxjYMKuUAC&pg=PA168">google.com</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Rotationsk%C3%B6rper&rft.au=A.+K.%26%2332%3BSharma&rft.btitle=Application+Of+Integral+Calculus&rft.date=2005&rft.genre=book&rft.isbn=8171419674&rft.pages=168&rft.pub=Discovery+Publishing+House" style="display:none"> </span></span></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><span class="book">Ravish R. Singh: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Engineering Mathematics</cite>. 6th Auflage. Tata McGraw-Hill, 1993, ISBN 0-07-014615-2, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>6.90</span> (englisch, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=oQ1y1HCpeowC&pg=SA6-PA90">google.com</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Rotationsk%C3%B6rper&rft.au=Ravish+R.%26%2332%3BSingh&rft.btitle=Engineering+Mathematics&rft.date=1993&rft.edition=6th&rft.genre=book&rft.isbn=0070146152&rft.pages=6.90&rft.pub=Tata+McGraw-Hill" style="display:none"> </span></span></span>

</li>

</ol>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://portal.dnb.de/opac.htm?method=simpleSearch&query=4136951-8">Literatur über Rotationskörper</a> im Katalog der <a href="Deutsche_Nationalbibliothek" title="Deutsche Nationalbibliothek">Deutschen Nationalbibliothek</a></li>

<li>Ronny Harbich: <style data-mw-deduplicate="TemplateStyles:r261891140">

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.mw-parser-output .webarchiv-memento a{color:inherit}

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</style><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110315155912/http://www.uni-magdeburg.de/harbich/rotationskoerper.php"><i>Rotationskörper.</i></a> (<span class="webarchiv-memento"><a href="Webarchivierung#Begrifflichkeiten" title="Webarchivierung">Memento</a></span> vom 15. März 2011 im <i><a href="Internet_Archive" title="Internet Archive">Internet Archive</a></i>). Bei: <i>Uni-Magdeburg.de.</i> 2003 (PDF; 948 kB).</li></ul>

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