{"id":204,"date":"2019-08-27T21:27:28","date_gmt":"2019-08-27T21:27:28","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=204"},"modified":"2020-01-15T20:49:13","modified_gmt":"2020-01-15T20:49:13","slug":"volume","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/volume\/","title":{"rendered":"Volume"},"content":{"rendered":"\n<script type=\"text\/x-mathjax-config\">\n  MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], [\"\\(\",\"\\)\"]] } });\n<\/script>\n\n\n\n<script type=\"text\/javascript\" src=\"http:\/\/cdn.mathjax.org\/mathjax\/latest\/MathJax.js?config=TeX-AMS_HTML\">\n<\/script>\n\n\n\n<meta http-equiv=\"X-UA-Compatible\" content=\"IE=EmulateIE7\">\n\n\n\n<title>Volume &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>Many three-dimensional solids can be generated by revolving a curve about the $x$-axis or $y$-axis.  For example, if we revolve the semi-circle given by $f(x)=\\sqrt{r^2-x^2}$ about the $x$-axis, we obtain a sphere of radius $r$.  We can derive the familiar formula for the volume of this sphere. <\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Finding the Volume of a Sphere<\/h4>\n\n\n\n<p>\n\nConsider a cross-section of the sphere as shown.  It is a circle with\nradius $f(x)$ and area $\\pi [f(x)]^2$.  Informally speaking, if we\n&#8220;slice&#8221; the sphere vertically into discs, each disc having\ninfinitesimal thickness $dx$, the volume of each disc is approximately\n$\\pi [f(x)]^2\\, dx$.  If we &#8220;add up&#8221; the volumes of the discs, we\nwill get the volume of the sphere:\n\\begin{eqnarray*}\n\tV&amp;=&amp;\\int^r_{-r} \\pi [f(x)]^2\\, dx\\\\\n\t&amp;=&amp; \\int^r_{-r} \\pi (r^2-x^2)\\, dx\\\\\n\t&amp;=&amp; \\left.\\pi \\left(r^2x-\\frac{x^3}{3}\\right)\\right|^r_{-r}\\\\\n\t&amp;=&amp; \\pi \\left(\\frac{2}{3}r^3\\right)-\\pi \\left(-\\frac{2}{3}r^3\\right)\\\\\n\t&amp;=&amp; \\frac{4}{3}\\pi r^3,\\quad{\\small\\textrm{as expected.}}\n\\end{eqnarray*}\nThis is called the <b>Method of Discs<\/b>.  In general, suppose\n$y=f(x)$ is nonnegative and continuous on $[a,b]$.  If the region\nbounded above by the graph of $f$, below by the $x$-axis, and on the\nsides by $x=a$ and $x=b$ is revolved about the $x$-axis, the volume\n$V$ of the generated solid is given by\n\\[V=\\int^a_b \\pi [f(x)]^2\\, dx.\\]\nWe can also obtain solids by revolving curves about the $y$-axis.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Revolving a Region about the $y$-axis<\/h4>\n\n\n\n<p>\n\nIf we revolve the region enclosed by $y=x^2$ and $y=2x$, $0\\leq x\\leq\n2$, about the $y$-axis, we generate a three-dimensional solid.\n\n<\/p>\n\n\n\n<p> Let&#8217;s find the volume of this solid.  If we &#8220;slice&#8221; the solid horizontally, each slice is a &#8220;washer.&#8221;  The outer radius is $\\sqrt{y}$, since $y=x^2 \\rightarrow x=\\sqrt{y}$, and the inner radius is $y\/2$, since $y=2x \\rightarrow x=y\/2$, and the thickness is $dy$.  The volume of each washer is therefore  \\[ [\\pi (\\sqrt{y})^2-\\pi (y\/2)^2]\\, dy.\\] Then the volume of the entire solid is given by \\begin{eqnarray*} \\int^4_0 [\\pi (\\sqrt{y})^2-\\pi (y\/2)^2]\\, dy&amp;=&amp;\\int^4_0 \\pi \\left[y-\\frac{y^2}{4}\\right]\\, dy\\\\ &amp;=&amp; \\left.\\pi \\left[\\frac{y^2}{2}-\\frac{y^3}{12}\\right]\\right|^4_0\\\\ &amp;=&amp; \\pi \\left( 8-\\frac{16}{3}\\right)-\\pi \\left(0-0\\right)\\\\ &amp;=&amp;\\frac{8\\pi}{3}. \\end{eqnarray*} This generalization of the Method of Discs is called the <b>Method of  Washers<\/b>.  As we have seen, these methods may be used when a region is revolved about either axis. <\/p>\n\n\n\n<p>\nWe could have taken a different approach in the previous example:\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Another Method<\/h4>\n\n\n\n<p> Look again at the volume of the solid generated by revolving the region enclosed by $y=2x$, $y=x^2$, $0\\leq x\\leq 2$ about the $y$-axis.  This time, we will view the solid as being composed of a collection of concentric cylindrical shells of radius $x$, height $2x-x^2$, and infinitesimal thickness $dx$.  The volume of each shell is approximately given by the lateral surface area $2\\pi\\cdot {\\small\\textrm{radius}}\\cdot {\\small\\textrm{height}}$ multiplied by the thickness: \\[2\\pi x[2x-x^2]\\, dx.\\] &#8220;Adding up&#8221; the volumes of the cylindrical shells, \\begin{eqnarray*} V&amp;=&amp; \\int^2_0 2\\pi x[2x-x^2]\\, dx\\\\ &amp;=&amp; \\int^2_0 2\\pi [2x^2-x^3]\\, dx\\\\ &amp;=&amp; \\left.\\left(\\frac{4}{3}\\pi x^3-\\frac{1}{2}\\pi x^4\\right)\\right|^2_0\\\\ &amp;=&amp; \\left(\\frac{32}{3}\\pi-8\\pi\\right)-\\left(0-0\\right)\\\\ &amp;=&amp; \\frac{8\\pi}{3},\\quad{\\small\\textrm{as found earlier.}} \\end{eqnarray*} This is called the <b>Method of Cylindrical Shells<\/b>.  Suppose $f(x)$, $g(x)$, $F(y)$, $G(y)$ satisfy all the requirements given earlier.  Then, for a region revolved about the $y$-axis, \\[V=\\int_a^b 2\\pi xf(x)\\, dx \\qquad{\\small\\textrm{or}}\\qquad V=\\int_a^b 2\\pi x[f(x)-g(x)]\\, dx.\\] For a region revolved about the $x$-axis, \\[V=\\int_c^d 2\\pi yF(y)\\, dy \\qquad{\\small\\textrm{or}}\\qquad V=\\int_c^d 2\\pi y[F(y)-G(y)]\\, dy.\\] <\/p>\n\n\n\n<center>\n<h4>Notes<\/h4>\n<\/center>\n\n\n\n<ul class=\"wp-block-list\"><li>  In the disc and washer methods, you integrate with respect to\n\t\tthe <i>same<\/i> variable as the axis about which you revolved the\n\t\tregion.\n\t<br><br>\n\t<\/li><li> In the method of cylindrical shells, you integrate with respect\n\t\tto the <i>other<\/i> variable.\n<\/li><\/ul>\n\n\n\n<p>\nComputing volumes using these methods takes some practice.  With\nexperience, you will be better able to visualize the solids and\ndetermine which method to apply.\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<center>\n<p>\n<\/p><h4>Key Concepts<\/h4>\n<p>\n<\/p><\/center>\n\n\n\n<b>Method of Washers:<\/b>\n\n\n\n<p>\n\\begin{eqnarray*}\n\tV = \\int^b_a \\pi ([f(x)]^2 &#8211; [g(x)]^2)\\, dx &amp; \\qquad{\\small\\textrm{or}}\\qquad &amp;\n\tV = \\int^d_c \\pi ([F(y)]^2 &#8211; [G(y)]^2)\\, dy. \\\\\n\\end{eqnarray*}\n\n<\/p>\n\n\n\n<p>\n<b>Method of Cylindrical Shells:<\/b>\n\\begin{eqnarray*}\n\tV = \\int^b_a 2\\pi xf(x)\\, dx &amp; \\qquad{\\small\\textrm{or}}\\qquad &amp;\n\tV = \\int^b_a 2\\pi x[f(x)-g(x)]\\, dx. \\\\\n\tV = \\int^d_c 2\\pi yF(y)\\, dy &amp; \\qquad{\\small\\textrm{or}}\\qquad &amp;\n\tV = \\int^d_c 2\\pi y[F(y)-G(y)]\\, dy. \\\\\n\\end{eqnarray*}\n\n<!------------------------>\n\n\n<br>\n\n<\/p>\n\n\n\n<hr class=\"wp-block-separator\"\/>\n\n\n\n<p>\n\n[<a href=\"https:\/\/physics.hmc.edu\/ct\/quiz\/QZ1410\/\">I&#8217;m ready to take the quiz.<\/a>]\n[<a href=\"#top\">I need to review more.<\/a>]<br>\n\n\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Volume &#8211; HMC Calculus Tutorial Many three-dimensional solids can be generated by revolving a curve about the $x$-axis or $y$-axis. For example, if we revolve the semi-circle given by $f(x)=\\sqrt{r^2-x^2}$ about the $x$-axis, we obtain a sphere of radius $r$. We can derive the familiar formula for the volume of this sphere. Finding the Volume&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":57,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-204","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/204","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/comments?post=204"}],"version-history":[{"count":4,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/204\/revisions"}],"predecessor-version":[{"id":1160,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/204\/revisions\/1160"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/57"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=204"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=204"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}