{"id":706,"date":"2025-10-09T15:33:13","date_gmt":"2025-10-09T15:33:13","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=706"},"modified":"2026-05-26T15:06:10","modified_gmt":"2026-05-26T15:06:10","slug":"triple-integrals-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/triple-integrals-learn-it-1\/","title":{"raw":"Triple Integrals: Learn It 1","rendered":"Triple Integrals: Learn It 1"},"content":{"raw":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Evaluate triple integrals over box-shaped regions<\/li>\r\n \t<li>Calculate triple integrals over more complex 3D regions<\/li>\r\n \t<li>Simplify a calculation by changing the order of integration of a triple integral<\/li>\r\n \t<li>Find the average value of a function of three variables<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2 data-type=\"title\">Integrable Functions of Three Variables<\/h2>\r\nIn earlier sections, we discussed the double integral of a function [latex]f(x,y)[\/latex] of two variables over a rectangular region in the plane. In this section we define the triple integral of a function [latex]f(x,y,z)[\/latex] of three variables over a rectangular solid box in space, [latex]\\mathbb{R}^{3}[\/latex].\r\n\r\nWe can define a rectangular box [latex]B[\/latex] in [latex]\\mathbb{R}^3[\/latex] as [latex]B = {(x,y,z)\\mid a \\leq x \\leq b, c \\leq y \\leq d, e \\leq z \\leq f }[\/latex]. We follow a similar procedure to what we did for double integrals over rectangular regions.\r\n\r\nWe divide the interval [latex][a,b][\/latex] into [latex]l[\/latex] subintervals [latex][x_{i-1},x_i][\/latex] of equal length [latex]\\Delta x = \\frac{x_i-x_{i-1}}{l}[\/latex], divide the interval [latex][c,d][\/latex] into [latex]m[\/latex] subintervals [latex][y_{j-1},y_j][\/latex] of equal length [latex]\\Delta y = \\frac{y_j-y_{j-1}}{m}[\/latex], and divide the interval [latex][e,f][\/latex] into [latex]n[\/latex] subintervals [latex][z_{k-1},z_k][\/latex] of equal length [latex]\\Delta z = \\frac{z_k-z_{k-1}}{n}[\/latex]. Then the rectangular box [latex]B[\/latex] is subdivided into [latex]lmn[\/latex] subboxes [latex]B_{ijk} = [x_{i-1},x_i] \\times [y_{j-1},y_j] \\times [z_{k-1},z_k][\/latex], as shown in Figure 1.\r\n\r\n[caption id=\"attachment_1372\" align=\"aligncenter\" width=\"590\"]<img class=\"size-full wp-image-1372\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144228\/5-4-1.jpeg\" alt=\"In x y z space, there is a box B with a subbox Bijk with sides of length Delta x, Delta y, and Delta z.\" width=\"590\" height=\"724\" \/> Figure 1.\u00a0A rectangular box in\u00a0[latex]\\mathbb{R}^{3}[\/latex] divided into subboxes by planes parallel to the coordinate planes.[\/caption]\r\n<p class=\"whitespace-normal break-words\">For each [latex]i[\/latex], [latex]j[\/latex], and [latex]k[\/latex], consider a sample point [latex]{({{x}^{*}_{ijk}},{{y}^{*}_{ijk}},{{z}^{*}_{ijk}})}[\/latex] in each sub-box [latex]B_{ijk}[\/latex]. We see that its volume is [latex]\\Delta V = \\Delta x\\Delta y\\Delta z[\/latex]. Form the triple Riemann sum:<\/p>\r\n<p style=\"text-align: center;\" data-start=\"66\" data-end=\"263\">[latex]{\\displaystyle\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}}{f}{({{x}^{*}_{ijk}},{{y}^{*}_{ijk}},{{z}^{*}_{ijk}})}{{\\Delta}{x}}{{\\Delta}{y}}{{\\Delta}{z}}[\/latex]<\/p>\r\n<p class=\"whitespace-normal break-words\">We define the triple integral in terms of the limit of a triple Riemann sum, as we did for the double integral in terms of a double Riemann sum.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>triple integral<\/h3>\r\n<p id=\"fs-id1167793943883\">The\u00a0<strong><span id=\"d140d1cd-af53-466a-8e69-402de1b91259_term219\" data-type=\"term\">triple integral<\/span><\/strong> of a function [latex]f(x, y, z)[\/latex] over a rectangular box [latex]B[\/latex] is defined as:<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\displaystyle\\lim_{{l,m,n}{\\rightarrow}{\\infty}}} \\ {\\displaystyle\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}} \\ {f}{({{x}^{*}_{ijk}},{{y}^{*}_{ijk}},{{z}^{*}_{ijk}})}{{\\Delta}{x}}{{\\Delta}{y}}{{\\Delta}{z}} = \\underset{B}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV}[\/latex]<\/p>\r\n<span style=\"font-size: 1rem; text-align: initial;\">if this limit exists.<\/span>\r\n\r\n<\/section>\r\n<p class=\"whitespace-normal break-words\">When the triple integral exists on [latex]B[\/latex], the function [latex]f(x, y, z)[\/latex] is said to be <strong>integrable<\/strong> on [latex]B[\/latex]. The triple integral exists if [latex]f(x, y, z)[\/latex] is continuous on [latex]B[\/latex], so we will use continuous functions for our examples. However, continuity is sufficient but not necessary; in other words, [latex]f[\/latex] is bounded on [latex]B[\/latex] and continuous except possibly on the boundary of [latex]B[\/latex] The sample point [latex]{({{x}^{*}_{ijk}},{{y}^{*}_{ijk}},{{z}^{*}_{ijk}})}[\/latex] can be any point in the rectangular sub-box [latex]{B}_{ijk}[\/latex] and all the properties of a double integral apply to a triple integral.<\/p>\r\n<p class=\"whitespace-normal break-words\">Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as with double integrals, we can use an iterated triple integral. Fubini's theorem extends to triple integrals as well.<\/p>\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>Fubini's theorem for triple integrals<\/h3>\r\n<p id=\"fs-id1167794145580\">If [latex]f(x, y, z)[\/latex] is continuous on a rectangular box [latex]{B} = {[a,b]} \\ {\\times} \\ {[c,d]} \\ {\\times} \\ {[e,f]}[\/latex], then:<\/p>\r\n\r\n<article class=\"text-token-text-primary w-full focus:outline-none [--shadow-height:45px] has-data-writing-block:pointer-events-none has-data-writing-block:-mt-(--shadow-height) has-data-writing-block:pt-(--shadow-height) [&amp;:has([data-writing-block])&gt;*]:pointer-events-auto [content-visibility:auto] supports-[content-visibility:auto]:[contain-intrinsic-size:auto_100lvh] scroll-mt-[calc(var(--header-height)+min(200px,max(70px,20svh)))]\" dir=\"auto\" tabindex=\"-1\" data-turn-id=\"request-WEB:2d67570c-a925-468f-a4db-d8502e6162c9-1\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"true\" data-turn=\"assistant\">\r\n<div class=\"text-base my-auto mx-auto pb-10 [--thread-content-margin:--spacing(4)] thread-sm:[--thread-content-margin:--spacing(6)] thread-lg:[--thread-content-margin:--spacing(16)] px-(--thread-content-margin)\">\r\n<div class=\"[--thread-content-max-width:40rem] thread-lg:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group\/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn\" tabindex=\"-1\">\r\n<div class=\"flex max-w-full flex-col grow\">\r\n<div class=\"min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal [.text-message+&amp;]:mt-1\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"71d96d09-2f69-4d6e-bda0-73871e23afc4\" data-message-model-slug=\"gpt-5\">\r\n<div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[1px]\">\r\n<div class=\"markdown prose dark:prose-invert w-full break-words dark markdown-new-styling\">\r\n<p style=\"text-align: center;\" data-start=\"49\" data-end=\"229\" data-is-last-node=\"\" data-is-only-node=\"\">[latex]{\\underset{B}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = {\\displaystyle\\int_{e}^{f}}{\\displaystyle\\int_{c}^{d}}{\\displaystyle\\int_{a}^{b}}{f}{(x,y,z)}{dx} \\ {dy} \\ {dz}}[\/latex]<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/article>This integral is also equal to any of the other five possible orderings for the iterated triple integral.\r\n\r\n<\/section>\r\n<p id=\"fs-id1167793263456\">For [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex], [latex]d[\/latex], [latex]e[\/latex], and [latex]f[\/latex] real numbers, the iterated triple integral can be expressed in six different orderings:<\/p>\r\n\r\n<center>[latex]\r\n\\begin{align}\r\n{\\displaystyle\\int_{e}^{f}}{\\displaystyle\\int_{c}^{d}}{\\displaystyle\\int_{a}^{b}}{f}{(x,y,z)}{dx} \\ {dy} \\ {dz} &amp;= {\\displaystyle\\int_{e}^{f}}\\left({\\displaystyle\\int_{c}^{d}}\\left({\\displaystyle\\int_{a}^{b}}{f}{(x,y,z)}{dx}\\right){dy}\\right){dz} = {\\displaystyle\\int_{c}^{d}}\\left({\\displaystyle\\int_{e}^{f}}\\left({\\displaystyle\\int_{a}^{b}}{f}{(x,y,z)}{dx}\\right){dz}\\right){dy} \\\\\r\n&amp;= {\\displaystyle\\int_{a}^{b}}\\left({\\displaystyle\\int_{e}^{f}}\\left({\\displaystyle\\int_{c}^{d}}{f}{(x,y,z)}{dy}\\right){dz}\\right){dx} = {\\displaystyle\\int_{e}^{f}}\\left({\\displaystyle\\int_{a}^{b}}\\left({\\displaystyle\\int_{c}^{d}}{f}{(x,y,z)}{dy}\\right){dx}\\right){dz} \\\\\r\n&amp;= {\\displaystyle\\int_{c}^{e}}\\left({\\displaystyle\\int_{a}^{b}}\\left({\\displaystyle\\int_{e}^{f}}{f}{(x,y,z)}{dz}\\right){dx}\\right){dy} = {\\displaystyle\\int_{a}^{b}}\\left({\\displaystyle\\int_{c}^{e}}\\left({\\displaystyle\\int_{e}^{f}}{f}{(x,y,z)}{dz}\\right){dy}\\right){dx}.\r\n\\end{align}\r\n[\/latex]<\/center>For a rectangular box, the order of integration does not make any significant difference in the level of difficulty in computation. We compute triple integrals using Fubini\u2019s Theorem rather than using the Riemann sum definition. We follow the order of integration in the same way as we did for double integrals (that is, from inside to outside).\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the following triple integral:<center>[latex]\\int_{z=0}^{z=1} \\int_{y=2}^{y=4} \\int_{x=-1}^{x=5} (x + yz^2)dxdydz[\/latex]<\/center>.[reveal-answer q=\"390606\"]Show Solution[\/reveal-answer][hidden-answer a=\"390606\"]The order of integration is specified in the problem, so integrate with respect to [latex]x[\/latex] first, then [latex]y[\/latex], and then [latex]z[\/latex].\r\n<center>[latex]\r\n\\begin{array}{lcl}\r\n&amp;\\displaystyle\\int_{z=0}^{z=1}\\int_{y=2}^{y=4}\\int_{x=-1}^{x=5}(x+yz^2)dx\\,dy\\,dz \\\\\r\n=&amp;\\displaystyle\\int_{z=0}^{z=1}\\int_{y=2}^{y=4}\\left[\\frac{x^2}{2}+xyz^2\\right]_{x=-1}^{x=5}dy\\,dz &amp;\\quad&amp; \\text{Integrate with respect to }x. \\\\\r\n=&amp;\\displaystyle\\int_{z=0}^{z=1}\\int_{y=2}^{y=4}[12+6yz^2]dy\\,dz &amp;\\quad&amp; \\text{Evaluate.} \\\\[0.5em]\r\n=&amp;\\displaystyle\\int_{z=0}^{z=1}\\left[12y+6\\frac{y^2 z^2}{2}\\right]_{y=2}^{y=4}dz &amp;\\quad&amp; \\text{Integrate with respect to }y. \\\\\r\n=&amp;\\displaystyle\\int_{z=0}^{z=1}[24+36z^2]dz &amp;\\quad&amp; \\text{Evaluate.} \\\\\r\n=&amp;\\left[24z+36\\frac{z^3}{3}\\right]_{z=0}^{z=1}=36.\r\n\\end{array}\r\n[\/latex]<\/center>[\/hidden-answer]<\/section><section class=\"textbox example\" aria-label=\"Example\">Evaluate the triple integral [latex]\\underset{B}{\\displaystyle\\iiint}{x^2}{yz}{dV}[\/latex] where[latex]{B} = {\\left \\{{(x,y,z)}{\\mid}{-2} \\ {\\leq} \\ {x} \\ {\\leq} \\ {1,0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {3,1} \\ {\\leq} \\ {z} \\ {\\leq} \\ {5} \\right \\}}[\/latex] as shown in the following figure.\r\n\r\n[caption id=\"attachment_1374\" align=\"aligncenter\" width=\"375\"]<img class=\"size-full wp-image-1374\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144322\/5-4-2.jpeg\" alt=\"In x y z space, there is a box given with corners (1, 0, 5), (1, 0, 1), (1, 3, 1), (1, 3, 5), (negative 2, 0, 5), (negative 2, 0, 1), (negative 2, 3, 1), and (negative 2, 3, 5).\" width=\"375\" height=\"328\" \/> Figure 2.\u00a0Evaluating a triple integral over a given rectangular box.[\/caption]\r\n\r\n[reveal-answer q=\"940127\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"940127\"]\r\n<p id=\"fs-id1167793937669\">The order is not specified, but we can use the iterated integral in any order without changing the level of difficulty. Choose, say, to integrate [latex]y[\/latex] first, then\u00a0[latex]x[\/latex], and then\u00a0[latex]z[\/latex].<\/p>\r\n[latex]\\hspace{2cm}\\begin{align}\r\n\r\n\\underset{B}{\\displaystyle\\iiint}{x^2}{yz}{dV}&amp;=\\displaystyle\\int_1^5\\displaystyle\\int_{-2}^1\\displaystyle\\int_0^3[x^2yz]dy \\ dx \\ dz =\\displaystyle\\int_1^5\\displaystyle\\int_{-2}^1\\left[x^2\\frac{y^2}2z\\bigg|_0^3\\right]dx \\ dz \\\\\r\n\r\n&amp;=\\displaystyle\\int_1^5\\displaystyle\\int_{-2}^1\\frac92x^2z \\ dx \\ dz = \\displaystyle\\int_1^5\\left[\\frac92\\frac{x^3}3z\\bigg|_{-2}^1\\right]dz=\\displaystyle\\int_1^5\\frac{27}2\\frac{z^2}2\\bigg|_1^5=162.\r\n\r\n\\end{align}[\/latex]\r\n\r\nNow try to integrate in a different order just to see that we get the same answer. Choose to integrate with respect to [latex]x[\/latex] first, then [latex]z[\/latex], and then [latex]y[\/latex].\r\n\r\n[latex]\\hspace{2cm}\\begin{align}\r\n\r\n\\underset{B}{\\displaystyle\\iiint}{x^2}{yz}{dV}&amp;=\\displaystyle\\int_0^3\\displaystyle\\int_1^5\\displaystyle\\int_{-2}^1[x^2yz]dx \\ dz \\ dy =\\displaystyle\\int_0^3\\displaystyle\\int_1^5\\left[\\frac{x^3}3yz\\bigg|_{-2}^1\\right]dz \\ dy \\\\\r\n\r\n&amp;=\\displaystyle\\int_0^3\\displaystyle\\int_1^53yz \\ dz \\ dy =\\displaystyle\\int_0^3\\left[3y\\frac{z^2}2\\bigg|_1^5\\right]dy=\\displaystyle\\int_0^336y \\ dy = 36\\frac{y^2}2\\bigg|_0^3=18(9-0)=162.\r\n\r\n\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]325847[\/ohm_question]<\/section>","rendered":"<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Evaluate triple integrals over box-shaped regions<\/li>\n<li>Calculate triple integrals over more complex 3D regions<\/li>\n<li>Simplify a calculation by changing the order of integration of a triple integral<\/li>\n<li>Find the average value of a function of three variables<\/li>\n<\/ul>\n<\/section>\n<h2 data-type=\"title\">Integrable Functions of Three Variables<\/h2>\n<p>In earlier sections, we discussed the double integral of a function [latex]f(x,y)[\/latex] of two variables over a rectangular region in the plane. In this section we define the triple integral of a function [latex]f(x,y,z)[\/latex] of three variables over a rectangular solid box in space, [latex]\\mathbb{R}^{3}[\/latex].<\/p>\n<p>We can define a rectangular box [latex]B[\/latex] in [latex]\\mathbb{R}^3[\/latex] as [latex]B = {(x,y,z)\\mid a \\leq x \\leq b, c \\leq y \\leq d, e \\leq z \\leq f }[\/latex]. We follow a similar procedure to what we did for double integrals over rectangular regions.<\/p>\n<p>We divide the interval [latex][a,b][\/latex] into [latex]l[\/latex] subintervals [latex][x_{i-1},x_i][\/latex] of equal length [latex]\\Delta x = \\frac{x_i-x_{i-1}}{l}[\/latex], divide the interval [latex][c,d][\/latex] into [latex]m[\/latex] subintervals [latex][y_{j-1},y_j][\/latex] of equal length [latex]\\Delta y = \\frac{y_j-y_{j-1}}{m}[\/latex], and divide the interval [latex][e,f][\/latex] into [latex]n[\/latex] subintervals [latex][z_{k-1},z_k][\/latex] of equal length [latex]\\Delta z = \\frac{z_k-z_{k-1}}{n}[\/latex]. Then the rectangular box [latex]B[\/latex] is subdivided into [latex]lmn[\/latex] subboxes [latex]B_{ijk} = [x_{i-1},x_i] \\times [y_{j-1},y_j] \\times [z_{k-1},z_k][\/latex], as shown in Figure 1.<\/p>\n<figure id=\"attachment_1372\" aria-describedby=\"caption-attachment-1372\" style=\"width: 590px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1372\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144228\/5-4-1.jpeg\" alt=\"In x y z space, there is a box B with a subbox Bijk with sides of length Delta x, Delta y, and Delta z.\" width=\"590\" height=\"724\" \/><figcaption id=\"caption-attachment-1372\" class=\"wp-caption-text\">Figure 1.\u00a0A rectangular box in\u00a0[latex]\\mathbb{R}^{3}[\/latex] divided into subboxes by planes parallel to the coordinate planes.<\/figcaption><\/figure>\n<p class=\"whitespace-normal break-words\">For each [latex]i[\/latex], [latex]j[\/latex], and [latex]k[\/latex], consider a sample point [latex]{({{x}^{*}_{ijk}},{{y}^{*}_{ijk}},{{z}^{*}_{ijk}})}[\/latex] in each sub-box [latex]B_{ijk}[\/latex]. We see that its volume is [latex]\\Delta V = \\Delta x\\Delta y\\Delta z[\/latex]. Form the triple Riemann sum:<\/p>\n<p style=\"text-align: center;\" data-start=\"66\" data-end=\"263\">[latex]{\\displaystyle\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}}{f}{({{x}^{*}_{ijk}},{{y}^{*}_{ijk}},{{z}^{*}_{ijk}})}{{\\Delta}{x}}{{\\Delta}{y}}{{\\Delta}{z}}[\/latex]<\/p>\n<p class=\"whitespace-normal break-words\">We define the triple integral in terms of the limit of a triple Riemann sum, as we did for the double integral in terms of a double Riemann sum.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>triple integral<\/h3>\n<p id=\"fs-id1167793943883\">The\u00a0<strong><span id=\"d140d1cd-af53-466a-8e69-402de1b91259_term219\" data-type=\"term\">triple integral<\/span><\/strong> of a function [latex]f(x, y, z)[\/latex] over a rectangular box [latex]B[\/latex] is defined as:<\/p>\n<p style=\"text-align: center;\">[latex]{\\displaystyle\\lim_{{l,m,n}{\\rightarrow}{\\infty}}} \\ {\\displaystyle\\sum_{i=1}^{l}}{\\displaystyle\\sum_{j=1}^{m}}{\\displaystyle\\sum_{k=1}^{n}} \\ {f}{({{x}^{*}_{ijk}},{{y}^{*}_{ijk}},{{z}^{*}_{ijk}})}{{\\Delta}{x}}{{\\Delta}{y}}{{\\Delta}{z}} = \\underset{B}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV}[\/latex]<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">if this limit exists.<\/span><\/p>\n<\/section>\n<p class=\"whitespace-normal break-words\">When the triple integral exists on [latex]B[\/latex], the function [latex]f(x, y, z)[\/latex] is said to be <strong>integrable<\/strong> on [latex]B[\/latex]. The triple integral exists if [latex]f(x, y, z)[\/latex] is continuous on [latex]B[\/latex], so we will use continuous functions for our examples. However, continuity is sufficient but not necessary; in other words, [latex]f[\/latex] is bounded on [latex]B[\/latex] and continuous except possibly on the boundary of [latex]B[\/latex] The sample point [latex]{({{x}^{*}_{ijk}},{{y}^{*}_{ijk}},{{z}^{*}_{ijk}})}[\/latex] can be any point in the rectangular sub-box [latex]{B}_{ijk}[\/latex] and all the properties of a double integral apply to a triple integral.<\/p>\n<p class=\"whitespace-normal break-words\">Now that we have developed the concept of the triple integral, we need to know how to compute it. Just as with double integrals, we can use an iterated triple integral. Fubini&#8217;s theorem extends to triple integrals as well.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>Fubini&#8217;s theorem for triple integrals<\/h3>\n<p id=\"fs-id1167794145580\">If [latex]f(x, y, z)[\/latex] is continuous on a rectangular box [latex]{B} = {[a,b]} \\ {\\times} \\ {[c,d]} \\ {\\times} \\ {[e,f]}[\/latex], then:<\/p>\n<article class=\"text-token-text-primary w-full focus:outline-none [--shadow-height:45px] has-data-writing-block:pointer-events-none has-data-writing-block:-mt-(--shadow-height) has-data-writing-block:pt-(--shadow-height) [&amp;:has([data-writing-block])&gt;*]:pointer-events-auto [content-visibility:auto] supports-[content-visibility:auto]:[contain-intrinsic-size:auto_100lvh] scroll-mt-[calc(var(--header-height)+min(200px,max(70px,20svh)))]\" dir=\"auto\" tabindex=\"-1\" data-turn-id=\"request-WEB:2d67570c-a925-468f-a4db-d8502e6162c9-1\" data-testid=\"conversation-turn-4\" data-scroll-anchor=\"true\" data-turn=\"assistant\">\n<div class=\"text-base my-auto mx-auto pb-10 [--thread-content-margin:--spacing(4)] thread-sm:[--thread-content-margin:--spacing(6)] thread-lg:[--thread-content-margin:--spacing(16)] px-(--thread-content-margin)\">\n<div class=\"[--thread-content-max-width:40rem] thread-lg:[--thread-content-max-width:48rem] mx-auto max-w-(--thread-content-max-width) flex-1 group\/turn-messages focus-visible:outline-hidden relative flex w-full min-w-0 flex-col agent-turn\" tabindex=\"-1\">\n<div class=\"flex max-w-full flex-col grow\">\n<div class=\"min-h-8 text-message relative flex w-full flex-col items-end gap-2 text-start break-words whitespace-normal [.text-message+&amp;]:mt-1\" dir=\"auto\" data-message-author-role=\"assistant\" data-message-id=\"71d96d09-2f69-4d6e-bda0-73871e23afc4\" data-message-model-slug=\"gpt-5\">\n<div class=\"flex w-full flex-col gap-1 empty:hidden first:pt-[1px]\">\n<div class=\"markdown prose dark:prose-invert w-full break-words dark markdown-new-styling\">\n<p style=\"text-align: center;\" data-start=\"49\" data-end=\"229\" data-is-last-node=\"\" data-is-only-node=\"\">[latex]{\\underset{B}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = {\\displaystyle\\int_{e}^{f}}{\\displaystyle\\int_{c}^{d}}{\\displaystyle\\int_{a}^{b}}{f}{(x,y,z)}{dx} \\ {dy} \\ {dz}}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/article>\n<p>This integral is also equal to any of the other five possible orderings for the iterated triple integral.<\/p>\n<\/section>\n<p id=\"fs-id1167793263456\">For [latex]a[\/latex], [latex]b[\/latex], [latex]c[\/latex], [latex]d[\/latex], [latex]e[\/latex], and [latex]f[\/latex] real numbers, the iterated triple integral can be expressed in six different orderings:<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{align}  {\\displaystyle\\int_{e}^{f}}{\\displaystyle\\int_{c}^{d}}{\\displaystyle\\int_{a}^{b}}{f}{(x,y,z)}{dx} \\ {dy} \\ {dz} &= {\\displaystyle\\int_{e}^{f}}\\left({\\displaystyle\\int_{c}^{d}}\\left({\\displaystyle\\int_{a}^{b}}{f}{(x,y,z)}{dx}\\right){dy}\\right){dz} = {\\displaystyle\\int_{c}^{d}}\\left({\\displaystyle\\int_{e}^{f}}\\left({\\displaystyle\\int_{a}^{b}}{f}{(x,y,z)}{dx}\\right){dz}\\right){dy} \\\\  &= {\\displaystyle\\int_{a}^{b}}\\left({\\displaystyle\\int_{e}^{f}}\\left({\\displaystyle\\int_{c}^{d}}{f}{(x,y,z)}{dy}\\right){dz}\\right){dx} = {\\displaystyle\\int_{e}^{f}}\\left({\\displaystyle\\int_{a}^{b}}\\left({\\displaystyle\\int_{c}^{d}}{f}{(x,y,z)}{dy}\\right){dx}\\right){dz} \\\\  &= {\\displaystyle\\int_{c}^{e}}\\left({\\displaystyle\\int_{a}^{b}}\\left({\\displaystyle\\int_{e}^{f}}{f}{(x,y,z)}{dz}\\right){dx}\\right){dy} = {\\displaystyle\\int_{a}^{b}}\\left({\\displaystyle\\int_{c}^{e}}\\left({\\displaystyle\\int_{e}^{f}}{f}{(x,y,z)}{dz}\\right){dy}\\right){dx}.  \\end{align}[\/latex]<\/div>\n<p>For a rectangular box, the order of integration does not make any significant difference in the level of difficulty in computation. We compute triple integrals using Fubini\u2019s Theorem rather than using the Riemann sum definition. We follow the order of integration in the same way as we did for double integrals (that is, from inside to outside).<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the following triple integral:<\/p>\n<div style=\"text-align: center;\">[latex]\\int_{z=0}^{z=1} \\int_{y=2}^{y=4} \\int_{x=-1}^{x=5} (x + yz^2)dxdydz[\/latex]<\/div>\n<p>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q390606\">Show Solution<\/button><\/p>\n<div id=\"q390606\" class=\"hidden-answer\" style=\"display: none\">The order of integration is specified in the problem, so integrate with respect to [latex]x[\/latex] first, then [latex]y[\/latex], and then [latex]z[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lcl}  &\\displaystyle\\int_{z=0}^{z=1}\\int_{y=2}^{y=4}\\int_{x=-1}^{x=5}(x+yz^2)dx\\,dy\\,dz \\\\  =&\\displaystyle\\int_{z=0}^{z=1}\\int_{y=2}^{y=4}\\left[\\frac{x^2}{2}+xyz^2\\right]_{x=-1}^{x=5}dy\\,dz &\\quad& \\text{Integrate with respect to }x. \\\\  =&\\displaystyle\\int_{z=0}^{z=1}\\int_{y=2}^{y=4}[12+6yz^2]dy\\,dz &\\quad& \\text{Evaluate.} \\\\[0.5em]  =&\\displaystyle\\int_{z=0}^{z=1}\\left[12y+6\\frac{y^2 z^2}{2}\\right]_{y=2}^{y=4}dz &\\quad& \\text{Integrate with respect to }y. \\\\  =&\\displaystyle\\int_{z=0}^{z=1}[24+36z^2]dz &\\quad& \\text{Evaluate.} \\\\  =&\\left[24z+36\\frac{z^3}{3}\\right]_{z=0}^{z=1}=36.  \\end{array}[\/latex]<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the triple integral [latex]\\underset{B}{\\displaystyle\\iiint}{x^2}{yz}{dV}[\/latex] where[latex]{B} = {\\left \\{{(x,y,z)}{\\mid}{-2} \\ {\\leq} \\ {x} \\ {\\leq} \\ {1,0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {3,1} \\ {\\leq} \\ {z} \\ {\\leq} \\ {5} \\right \\}}[\/latex] as shown in the following figure.<\/p>\n<figure id=\"attachment_1374\" aria-describedby=\"caption-attachment-1374\" style=\"width: 375px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1374\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144322\/5-4-2.jpeg\" alt=\"In x y z space, there is a box given with corners (1, 0, 5), (1, 0, 1), (1, 3, 1), (1, 3, 5), (negative 2, 0, 5), (negative 2, 0, 1), (negative 2, 3, 1), and (negative 2, 3, 5).\" width=\"375\" height=\"328\" \/><figcaption id=\"caption-attachment-1374\" class=\"wp-caption-text\">Figure 2.\u00a0Evaluating a triple integral over a given rectangular box.<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q940127\">Show Answer<\/button><\/p>\n<div id=\"q940127\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793937669\">The order is not specified, but we can use the iterated integral in any order without changing the level of difficulty. Choose, say, to integrate [latex]y[\/latex] first, then\u00a0[latex]x[\/latex], and then\u00a0[latex]z[\/latex].<\/p>\n<p>[latex]\\hspace{2cm}\\begin{align}    \\underset{B}{\\displaystyle\\iiint}{x^2}{yz}{dV}&=\\displaystyle\\int_1^5\\displaystyle\\int_{-2}^1\\displaystyle\\int_0^3[x^2yz]dy \\ dx \\ dz =\\displaystyle\\int_1^5\\displaystyle\\int_{-2}^1\\left[x^2\\frac{y^2}2z\\bigg|_0^3\\right]dx \\ dz \\\\    &=\\displaystyle\\int_1^5\\displaystyle\\int_{-2}^1\\frac92x^2z \\ dx \\ dz = \\displaystyle\\int_1^5\\left[\\frac92\\frac{x^3}3z\\bigg|_{-2}^1\\right]dz=\\displaystyle\\int_1^5\\frac{27}2\\frac{z^2}2\\bigg|_1^5=162.    \\end{align}[\/latex]<\/p>\n<p>Now try to integrate in a different order just to see that we get the same answer. Choose to integrate with respect to [latex]x[\/latex] first, then [latex]z[\/latex], and then [latex]y[\/latex].<\/p>\n<p>[latex]\\hspace{2cm}\\begin{align}    \\underset{B}{\\displaystyle\\iiint}{x^2}{yz}{dV}&=\\displaystyle\\int_0^3\\displaystyle\\int_1^5\\displaystyle\\int_{-2}^1[x^2yz]dx \\ dz \\ dy =\\displaystyle\\int_0^3\\displaystyle\\int_1^5\\left[\\frac{x^3}3yz\\bigg|_{-2}^1\\right]dz \\ dy \\\\    &=\\displaystyle\\int_0^3\\displaystyle\\int_1^53yz \\ dz \\ dy =\\displaystyle\\int_0^3\\left[3y\\frac{z^2}2\\bigg|_1^5\\right]dy=\\displaystyle\\int_0^336y \\ dy = 36\\frac{y^2}2\\bigg|_0^3=18(9-0)=162.    \\end{align}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm325847\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=325847&theme=lumen&iframe_resize_id=ohm325847&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":5,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-706","chapter","type-chapter","status-publish","hentry"],"part":434,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/706","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":26,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/706\/revisions"}],"predecessor-version":[{"id":2890,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/706\/revisions\/2890"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/434"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/706\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=706"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=706"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=706"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=706"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}