ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Visual calculus</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Visual calculus</span></span>

</h1>

</header>

</p>

<p><b>Visual calculus</b>, invented by <a href="Mamikon_Mnatsakanian" title="Mamikon Mnatsakanian">Mamikon Mnatsakanian</a> (known as Mamikon), is an approach to solving a variety of <a href="Integral" title="Integral">integral calculus</a> problems.<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> Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation. Mamikon collaborated with <a href="Tom_Apostol" class="mw-redirect" title="Tom Apostol">Tom Apostol</a> on the 2013 book <i>New Horizons in Geometry</i> describing the subject.

</p>

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

<p>Mamikon devised his method in 1959 while an undergraduate, first applying it to a well-known geometry problem: find the area of a ring (<a href="Annulus_(mathematics)" title="Annulus (mathematics)">annulus</a>), given the length of a <a href="Chord_(geometry)" title="Chord (geometry)">chord</a> tangent to the inner circumference. Perhaps surprisingly, no additional information is needed; the solution does not depend on the ring's inner and outer dimensions.

</p><p>The traditional approach involves algebra and application of the <a href="Pythagorean_theorem" title="Pythagorean theorem">Pythagorean theorem</a>. Mamikon's method, however, envisions an alternate construction of the ring: first the inner circle alone is drawn, then a constant-length tangent is made to travel along its circumference, "sweeping out" the ring as it goes.

</p><p>Now if all the (constant-length) tangents used in constructing the ring are translated so that their points of tangency coincide, the result is a circular disk of known radius (and easily computed area). Indeed, since the inner circle's radius is irrelevant, one could just as well have started with a circle of radius zero (a point)—and sweeping out a ring around a circle of zero radius is indistinguishable from simply rotating a line segment about one of its endpoints and sweeping out a disk.

</p><p>Mamikon's insight was to recognize the equivalence of the two constructions; and because they are equivalent, they yield equal areas. Moreover, the two starting curves need not be circular—a finding not easily proven by more traditional geometric methods. This yields <b>Mamikon's theorem</b>:

</p>

<dl><dd><i>The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve.</i></dd></dl>

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

<div class="mw-heading mw-heading3"><h3 id="Area_of_a_cycloid">Area of a cycloid</h3></div>

<p>The area of a <a href="Cycloid" title="Cycloid">cycloid</a> can be calculated by considering the area between it and an enclosing rectangle. These tangents can all be clustered to form a circle. If the circle generating the cycloid has radius <span class="texhtml"><i>r</i></span> then this circle also has radius <span class="texhtml"><i>r</i></span> and area <span class="texhtml">π<i>r</i><sup>2</sup></span>. The area of the rectangle is <span class="texhtml">2<i>r</i> × 2π<i>r</i> = 4π<i>r</i><sup>2</sup></span>. Therefore, the area of the cycloid is <span class="texhtml">3π<i>r</i><sup>2</sup></span>: it is 3 times the area of the generating circle.

</p><p>The tangent cluster can be seen to be a circle because the cycloid is generated by a circle and the tangent to the cycloid will be at right angle to the line from the generating point to the rolling point. Thus the tangent and the line to the contact point form a right-angled triangle in the generating circle. This means that clustered together the tangents will describe the shape of the generating circle.<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>

<ul><li><a href="Cavalieri's_principle" title="Cavalieri's principle">Cavalieri's principle</a></li>

<li><a href="Hodograph" title="Hodograph">Hodograph</a> – This is a related construct that maps the velocity of a point using a polar diagram.</li>

<li><i><a href="The_Method_of_Mechanical_Theorems" title="The Method of Mechanical Theorems">The Method of Mechanical Theorems</a></i></li>

<li><a href="Pappus's_centroid_theorem" title="Pappus's centroid theorem">Pappus's centroid theorem</a></li>

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

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

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.cco.caltech.edu/~mamikon/calculus.html">Visual Calculus</a> Mamikon Mnatsakanian</span>

</li>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFHaunspergerKennedy2006" class="citation book cs1">Haunsperger, Deanna; Kennedy, Stephen (2006). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=I9oVP8TlyqIC&pg=PA70"><i>The Edge of the Universe: Celebrating Ten Years of Math Horizons</i></a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780883855553</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">May 9,</span> 2017</span>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFApostol,_Mnatsakanian2012" class="citation book cs1">Apostol, Mnatsakanian (2012). <i>New Horizons in Geometry</i>. Mathematical Association of America. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5948%2F9781614442103">10.5948/9781614442103</a> (inactive July 1, 2025). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781614442103</bdi>.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: DOI inactive as of July 2025 (link)</span></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://www.its.caltech.edu/~mamikon/">ProjMath Mamikon</a></li>

<li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/ProofwithoutWords.html">Proof without Words</a> from <a href="MathWorld" title="MathWorld">MathWorld</a></li>

<li><a rel="nofollow" class="external text" href="https://demonstrations.wolfram.com/MamikonsMethodForTheAreaOfTheCycloid/">Wolfram Interactive Demonstration of Mamikon's theorem</a></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-07-12" href="https://en.wikipedia.org/wiki/?title=Visual_calculus&oldid=1300152919">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>