{"id":210,"date":"2019-08-27T23:22:34","date_gmt":"2019-08-27T23:22:34","guid":{"rendered":"http:\/\/104.42.120.246.xip.io\/calculus-tutorials\/?page_id=210"},"modified":"2019-12-05T00:30:03","modified_gmt":"2019-12-05T00:30:03","slug":"multiple-integration","status":"publish","type":"page","link":"https:\/\/math.hmc.edu\/calculus\/hmc-mathematics-calculus-online-tutorials\/multivariable-calculus\/multiple-integration\/","title":{"rendered":"Multiple Integration"},"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>Multiple Integration &#8211; HMC Calculus Tutorial<\/title>\n\n\n\n<p>\n<!------------------------>\n\nRecall our definition of the definite integral of a function of a <i>single<\/i> variable:\n\n<\/p>\n\n\n\n<p>\n\t\tLet $f(x)$ be defined on $[a,b]$ and let $x_{0},x_{1},\\ldots,x_{n}$ be\n\t\ta partition of $[a,b]$.  For each $[x_{i-1},x_{i}]$, let $x_{i}^{*}\n\t\t\\in [x_{i-1},x_{i}]$.  Then\n\t\t$$\n\t\t\\int_{a}^{b}f(x)\\,dx = \\lim_{\\max \\Delta x_{i} \\rightarrow 0}\n\t\t\\sum_{i=1}^{n}f(x_{i}^{*})\\Delta x_{i}.\n\t\t$$\n<\/p>\n\n\n\n<p> Take a quick look at the <a href=\"https:\/\/math.hmc.edu\/calculus-tutorials\/hmc-mathematics-calculus-online-tutorials\/single-variable-calculus\/riemann-sums\/?preview_id=192&amp;preview_nonce=eb59e62529&amp;preview=true\">Riemann Sum Tutorial<\/a>. <\/p>\n\n\n\n<p>\nWe can extend this definition to define the integral of a function of\ntwo or more variables.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Double Integral of a Function of Two Variables<\/h4>\n\n\n\n<p> Let $f(x,y)$ be defined on a closed and bounded region $R$ of the $xy$-plane.  Set up a grid of vertical and horizontal lines in the $xy$-plane to form an <b>inner partition<\/b> of $R$ into $n$ rectangular subregions $R_{k}$ of area $\\Delta A_{k}$, each of which lies entirely in $R$.  Ignore the rectangles that are not entirely contained in $R$.  Choose a point $(x_{k}^{*}, y_{k}^{*})$ in each subregion $R_{k}$.  The sum $$ \\sum_{k=1}^{n} f(x_{k}^{*}, y_{k}^{*}) \\Delta A_{k} $$ is called a <b>Riemann Sum<\/b>.  In the limit as we make our grid more and more dense, we define the <b>double integral of $f(x,y)$ over $R$<\/b> as $$ \\iint\\limits_R f(x,y\\,)dA= \\lim_{\\max \\Delta A_{k} \\rightarrow 0} \\sum_{k=1}^{n} f(x_{k}^{*}, y_{k}^{*}) \\Delta A_{k} . $$ <\/p>\n\n\n\n<center>\n<p>\n<h4>Notes<\/h4>\n<\/p><p>\n<\/p><\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> If this limit exists, we say that $f$ is <b>integrable<\/b> over\n\t\tthe <b>region of integration<\/b> $R$.\n\t<br><br>\n\t<\/li><li> If $f$ is continuous on $R$, then $f$ is integrable over $R$.\n\n<\/li><\/ul>\n\n\n\n<h4 class=\"wp-block-heading\">Geometric Interpretation of the Double Integral<\/h4>\n\n\n\n<p>\n\nNotice that as we increase the density of our grid, the sum\n$\\sum\\limits_{k=1}^{n}A_{k}$ of the individual rectangles better and\nbetter approximates the area of region $R$.  In the limit as $\\Delta\nA_{k} \\rightarrow 0$, we have\n$$\n\\mbox{Area of } R = \\iint\\limits_{R} dA.\n$$\nSuppose now that $f(x,y) \\ge 0$ on $R$.  Then $f(x_{k}^{*},\ny_{k}^{*})\\Delta A_{k}$ is the volume of a rectangular parallelopiped\nof height $f(x_{k}^{*},y_{k}^{*})$ and base area $\\Delta A_k$.  Adding\nup these volumes, we get an appoximation for the volume of the solid\nabove $R$ and below the suface $z=f(x,y)$.  Thus, in the limit as\n$\\Delta A_{k} \\rightarrow 0$,\n<\/p>\n\n\n\n<center>\n<p>\n<h4>Note<\/h4>\n<\/p><p>\n<\/p><\/center>\n\n\n\n<p>\n\nThe interpretation of the double integral as a volume still holds if\n$f(x,y)$ takes on both positive and negative values.  In this case, we\nobtain the difference between the volumes <\/p>\n\n\n\n<i>above<\/i>\n\n\n\n<p> the $xy$-plane\nbetween $z=f(x,y)$ and $R$ and the volume <\/p>\n\n\n\n<i>below<\/i>\n\n\n\n<p> the $xy$-plane\nbetween $z=f(x,y)$ and $R$.\n\n<\/p>\n\n\n\n<p>\nWe next turn to the actual evaluation of double integrals.\n\n<\/p>\n\n\n\n<h4 class=\"wp-block-heading\">Iterated Integrals<\/h4>\n\n\n\n<p>\n\nIn the double integral $\\iint\\limits_{R} f(x,y) \\,dA$, the differential $dA$ may be\nviewed informally as an infinitesimal area of a rectangle inside $R$\nwith dimensions $dy$ and $dx$.  For the kinds of &#8220;ordinary&#8221;\nfunctions and regions we&#8217;ll be concerned with,\n\\begin{eqnarray*}\n\t\\iint\\limits_{R} f(x,y)dA &amp; = &amp; \\int_{a}^{b}\\left[  \n\t \\int_{g_{1}(x)}^{g_{2}(x)} f(x,y\\,)dy \\right] \\,dx =\n\t \\int_{a}^{b}\\int_{g_{1}(x)}^{g_{2}(x)} f(x,y\\,)dy\\,dx  \\\\\n\t &amp; = &amp; \\int_{c}^{d} \\left[ \\int_{h_{1}(y)}^{h_{2}(y)} f(x,y) \\,dx \\right]\n\t \\,dy =    \\int_{c}^{d}\\int_{h_{1}(y)}^{h_{2}(y)} f(x,y)\\, dx\\,dy\n\\end{eqnarray*}\n\n<\/p>\n\n\n\n<p>\nwhere the limits of integration are determined by the region $R$ over\nwhich we are integrating.\n\n<\/p>\n\n\n\n<center>\n<p>\n<h4>Notes<\/h4>\n<\/p><p>\n<\/p><\/center>\n\n\n\n<ul class=\"wp-block-list\"><li> These integrals are called <b>iterated integrals<\/b>, since we\n\t\tintegrate more than once.\n\t<br><br>\n\t<\/li><li> We integrate &#8220;from the inside out.&#8221;  That is, in\n\t\t$\\displaystyle\\int_{a}^{b}\\int_{g_{1}(x)}^{g_{2}(x)}\n\t\tf(x,y)\\,dy\\,dx$, we \n\t\tfirst integrate $f(x,y)$ with respect to $y$ and evaluate it at\n\t\t$g_{2}(x)$ and $g_{1}(x)$.  We then integrate the result with respect\n\t\tto $x$ and evaluate the outcome at $a$ and $b$.\n\t<br><br>\n\t<\/li><li> Iterated triple integrals $\\displaystyle\\iiint\\limits_{G} f(x,y,z)dV$\n\t\tcan be defined in a similar way.\n\n<\/li><\/ul>\n\n\n\n<p>\nAn example will make these ideas more concrete.\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nLet&#8217;s evaluate the double integral $\\displaystyle \\iint\\limits_{R}\n6xy\\, dA$, where $R$ is the region bounded by $y=0$, $x=2$, and\n$y=x^{2}$.  We will verify here that the order of integration is\nunimportant:\n\n<\/p>\n\n\n\n<p>\n\n\t\t\\begin{eqnarray*}\n\t\t\t\\iint\\limits_{R} 6xy dA &amp; = &amp; \\int_{0}^{2}\\int_{0}^{x^{2}} 6xy\\,\n\t\t\t dy\\,dx \\\\ \n\t\t\t &amp; = &amp; \\int_{0}^{2}\\left[ \\left. 3xy^{2} \\right|_{y=0}^{x^{2}} \\right]\\,\n\t\t\t dx \\\\\n\t\t\t&amp; = &amp; \\int_{0}^{2} 3x^{5}\\, dx\\\\\n\t\t\t&amp; = &amp; \\left. \\frac{1}{2} x^{6} \\right|_{x=0}^{2}\\\\\n\t\t\t&amp; = &amp;  \\frac{1}{2}(64)-\\frac{1}{2}(0)\\\\\n\t\t\t&amp; = &amp; 32\n\t\t\\end{eqnarray*}\n\n<\/p>\n\n\n\n<p>\n\n\t\t\\begin{eqnarray*}\n\t\t\t\\iint\\limits_{R} 6xy dA &amp; = &amp; \\int_{0}^{4}\\int_{\\sqrt{y}}^{2} 6xy\n\t\t\t \\,dx\\,dy \\\\\n\t\t\t &amp; = &amp; \\int_{0}^{4}\\left[ \\left. 3x^{2y} \\right|_{x=\\sqrt{y}}^{2}\n\t\t\t \\right] \\, dy \\\\\n\t\t\t&amp; = &amp;  \\int_{0}^{4}\\left( 12y-3y^{2} \\right) \\, dy \\\\\n\t\t\t&amp; = &amp; \\left. \\left( 6y^{2} -y^{3} \\right) \\right|^{4}_{y=0} \\\\\n\t\t\t&amp; = &amp; \\left( 6(4)^{2}-(4)^{3} \\right) &#8211; \\left( 6(0)^{2}- (0)^{3}\n\t\t\t \\right) \\\\\n\t\t\t&amp; = &amp; 32\n\t\t\\end{eqnarray*}\n\t\t<br><br>\n\n<\/p>\n\n\n\n<p>\nso $\\displaystyle \\iint\\limits_{R} 6xy \\, dA =32$ here,\nregardless of the order in which we carry out the integration, as long\nas we are careful to set up the limits of integration correctly.\n\n<\/p>\n\n\n\n<p>\nNow for a triple integral&#8230;\n\n<\/p>\n\n\n\n<h6 class=\"wp-block-heading\">Example<\/h6>\n\n\n\n<p>\n\nWe will evaluate the triple integral $\\displaystyle\n\\int_{0}^{2}\\int_{-1}^{y^{2}}\\int_{1}^{z} yz \\, dx\n\\,dz\\,dy.$\n\n<\/p>\n\n\n\n<p>\n\t\t\n\t\t\t\\begin{eqnarray*}\n\t\t\t\t\\int_{0}^{2}\\int_{-1}^{y^{2}}\\int_{1}^{z} yz \\, dx\n\t\t\t\t\\,dz\\,dy &amp; = &amp; \\int_{0}^{2}\\int_{-1}^{y^{2}}\n\t\t\t\t\\left[ \\left. (xyz) \\right|_{x\\,=\\,1}^{x\\,=\\,z} \\right]\\,dz\\,dy  \\\\\n\t\t\t\t &amp; = &amp;  \\int_{0}^{2}\\int_{-1}^{y^{2}} \\left( yz^{2}-yz\n\t\t\t\t \\right)\\,dz\\,dy \\\\\n\t\t\t\t&amp; = &amp; \\int_{0}^{2} \\left[ \\left. \\left\n\t\t\t\t( \\frac{yz^{3}}{3}-\\frac{yz^{2}}{2}\\right) \\right|_{z\\,=\\,-1}^{z\\,=\\,y^{2}}\n\t\t\t\t\\right] \\,dy\\\\\n\t\t\t\t&amp; = &amp; \\int_{0}^{2} \\left( \\frac{y^{7}}{3} &#8211; \\frac{y^{5}}{2} +\n\t\t\t\t\\frac{5y}{6} \\right)\\,dy\\\\\n\t\t\t\t&amp; = &amp; \\left. \\left(\\frac{y^{8}}{24}-\\frac{y^{6}}{12}+\\frac{5y^{2}}{12}\n\t\t\t\t\\right) \\right|_{0}^{2}\\\\\n\t\t\t\t&amp; = &amp; \\frac{264}{24}-\\frac{64}{12}+\\frac{20}{12}\\\\\n\t\t\t\t&amp; = &amp; \\frac{84}{12}\\\\\n\t\t\t\t&amp; = &amp; 7.\n\t\t\t\\end{eqnarray*} \n<\/p>\n\n\n\n<p>\n\t\t\t\tIntegrate with respect to $x$ first.\n<\/p>\n\n\n\n<p>\n\t\t\t\tNext integrate with respect to $z$.\n<\/p>\n\n\n\n<p>\n\t\t\t\tFinally, integrate with respect to $y$.\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<p>\n\nLet $f(x,y)$ be defined on a closed and bounded region $R$ of the\n$xy$-plane.  Then\n$$\n\\iint\\limits_{R} f(x,y) \\, dA = \\lim_{\\max \\Delta A_{k}\n\\rightarrow 0} \\sum_{k=1}^{n} f(x_{k}^{*}, y_{k}^{*}) \\Delta A_{k} \n$$\nwhere each $\\Delta A_{k}$ gives the area of a rectangle in an inner\npartition of $R$.\n\n<\/p>\n\n\n\n<p>\nWe evaluate $\\displaystyle\\iint\\limits_{R} f(x,y) \\, dA$\nas an iterated integral:\n\\begin{eqnarray*}\n\t\\iint\\limits_{R} f(x,y) \\, dA &amp; = &amp;\n\t \\int_{a}^{b}\\int_{g_{1}(x)}^{g_{2}(x)}\n\t f(x,y\\,)dy\\,dx  \\\\\n\t &amp; = &amp;  \\int_{c}^{d}\\int_{h_{1}(y)}^{h_{2}(y)} f(x,y)\\,\n\t dx\\,dy \n\\end{eqnarray*}\nfor &#8220;ordinary&#8221; regions $R$ and functions $f(x,y)$.\n\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\/QZ2510\/\">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>Multiple Integration &#8211; HMC Calculus Tutorial Recall our definition of the definite integral of a function of a single variable: Let $f(x)$ be defined on $[a,b]$ and let $x_{0},x_{1},\\ldots,x_{n}$ be a partition of $[a,b]$. For each $[x_{i-1},x_{i}]$, let $x_{i}^{*} \\in [x_{i-1},x_{i}]$. Then $$ \\int_{a}^{b}f(x)\\,dx = \\lim_{\\max \\Delta x_{i} \\rightarrow 0} \\sum_{i=1}^{n}f(x_{i}^{*})\\Delta x_{i}. $$ Take a&hellip;<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":59,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"tags":[],"class_list":["post-210","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/210","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=210"}],"version-history":[{"count":5,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/210\/revisions"}],"predecessor-version":[{"id":1125,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/210\/revisions\/1125"}],"up":[{"embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/pages\/59"}],"wp:attachment":[{"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/media?parent=210"}],"wp:term":[{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.hmc.edu\/calculus\/wp-json\/wp\/v2\/tags?post=210"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}