{"id":1068,"date":"2021-11-01T19:16:47","date_gmt":"2021-11-01T19:16:47","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1068"},"modified":"2022-11-01T04:28:19","modified_gmt":"2022-11-01T04:28:19","slug":"average-value-of-a-function-of-three-variables","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/average-value-of-a-function-of-three-variables\/","title":{"raw":"Average Value of a Function of Three Variables","rendered":"Average Value of a Function of Three Variables"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul class=\"os-abstract\">\r\n \t<li><span class=\"os-abstract-content\">Calculate the average value of a function of three variables.<\/span><\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-id1167793311689\">Recall that we found the average value of a function of two variables by evaluating the double integral over a region on the plane and then dividing by the area of the region. Similarly, we can find the average value of a function in three variables by evaluating the triple integral over a solid region and then dividing by the volume of the solid.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3 style=\"text-align: center;\">theorem: average value of a function of three variables<\/h3>\r\n\r\n<hr \/>\r\n<p id=\"fs-id1167793311703\">If [latex]f(x, y, z)[\/latex] is integrable over a solid bounded region [latex]E[\/latex] with positive volume [latex]V(E)[\/latex], then the average value of the function is<\/p>\r\n<p style=\"text-align: center;\">[latex]{f_{\\text{ave}}} = {\\frac{1}{V(E)}}\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV}[\/latex].<\/p>\r\n<p id=\"fs-id1167793422958\">Note that the volume is [latex]{V(E)} = \\underset{E}{\\displaystyle\\iiint}{1}{dV}[\/latex].<\/p>\r\n\r\n<\/div>\r\n<div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: finding an average temperature<\/h3>\r\nThe temperature at a point [latex](x, y, z)[\/latex] of a solid [latex]E[\/latex] bounded by the coordinate planes and the plane [latex]x+y+z=1[\/latex] is [latex]{T}{(x,y,z)} = {(xy+8z+20)}{^\\circ}{\\text{C}}[\/latex]. Find the average temperature over the solid.\r\n\r\n[reveal-answer q=\"067483257\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"067483257\"]\r\n<p id=\"fs-id1167793473246\">Use the theorem given above and the triple integral to find the numerator and the denominator. Then do the division. Notice that the plane [latex]x+y+z=1[\/latex] has intercepts\u00a0[latex](1, 0, 0)[\/latex], [latex](0, 1, 0)[\/latex], and [latex](0, 0, 1)[\/latex]. The region [latex]E[\/latex] looks like<\/p>\r\n<p style=\"text-align: center;\">[latex]{E} = {\\left \\{ {(x,y,z)}{\\mid}{0} \\ {\\leq} \\ {x} \\ {\\leq} \\ {1,0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {{1-x},{0}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {1-x-y} \\right \\}}[\/latex].<\/p>\r\n<p id=\"fs-id1167793630666\">Hence the triple integral of the temperature is<\/p>\r\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = {\\displaystyle\\int^{x=1}_{x=0}} \\ {\\displaystyle\\int^{y=1-x}_{y=0}} \\ {\\displaystyle\\int^{z=1-x-y}_{z=0}}(xy+8z+20){dz}{dy}{dx} = {\\frac{147}{40}}[\/latex].<\/p>\r\nThe volume evaluation is\r\n<p style=\"text-align: center;\">[latex]{V}{(E)} = \\underset{E}{\\displaystyle\\iiint}{1dV} = {\\displaystyle\\int^{x=1}_{x=0}} \\ {\\displaystyle\\int^{y=1-x}_{y=0}} \\ {\\displaystyle\\int^{z=1-x-y}_{z=0}}{1dzdydx} = {\\frac{1}{6}}[\/latex].<\/p>\r\n<p id=\"fs-id1167794160233\">Hence the average value is [latex]{{T}_{\\text{ave}}} = {\\frac{147\/40}{1\/6}} = {\\frac{6(147)}{40}} = {\\frac{441}{20}}[\/latex] degrees Celsius.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\nFind the average value of the function [latex]f(x, y, z)=xyz[\/latex] over the cube with sides of length 4 units in the first octant with one vertex at the origin and edges parallel to the coordinate axes.\r\n\r\n[reveal-answer q=\"151360489\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"151360489\"]\r\n\r\n[latex]f_{\\text{ave}}=8[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n[caption]Watch the following video to see the worked solution to the above Try It[\/caption]\r\n\r\n<center><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=8197107&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=6U0zv9wquVo&amp;video_target=tpm-plugin-7r1ilu60-6U0zv9wquVo\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/center><center>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP5.26_transcript.html\">transcript for \u201cCP 5.26\u201d here (opens in new window).<\/a><\/center><\/div>\r\n<section data-depth=\"1\"><\/section>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul class=\"os-abstract\">\n<li><span class=\"os-abstract-content\">Calculate the average value of a function of three variables.<\/span><\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-id1167793311689\">Recall that we found the average value of a function of two variables by evaluating the double integral over a region on the plane and then dividing by the area of the region. Similarly, we can find the average value of a function in three variables by evaluating the triple integral over a solid region and then dividing by the volume of the solid.<\/p>\n<div class=\"textbox shaded\">\n<h3 style=\"text-align: center;\">theorem: average value of a function of three variables<\/h3>\n<hr \/>\n<p id=\"fs-id1167793311703\">If [latex]f(x, y, z)[\/latex] is integrable over a solid bounded region [latex]E[\/latex] with positive volume [latex]V(E)[\/latex], then the average value of the function is<\/p>\n<p style=\"text-align: center;\">[latex]{f_{\\text{ave}}} = {\\frac{1}{V(E)}}\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV}[\/latex].<\/p>\n<p id=\"fs-id1167793422958\">Note that the volume is [latex]{V(E)} = \\underset{E}{\\displaystyle\\iiint}{1}{dV}[\/latex].<\/p>\n<\/div>\n<div>\n<div class=\"textbox exercises\">\n<h3>Example: finding an average temperature<\/h3>\n<p>The temperature at a point [latex](x, y, z)[\/latex] of a solid [latex]E[\/latex] bounded by the coordinate planes and the plane [latex]x+y+z=1[\/latex] is [latex]{T}{(x,y,z)} = {(xy+8z+20)}{^\\circ}{\\text{C}}[\/latex]. Find the average temperature over the solid.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q067483257\">Show Solution<\/span><\/p>\n<div id=\"q067483257\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167793473246\">Use the theorem given above and the triple integral to find the numerator and the denominator. Then do the division. Notice that the plane [latex]x+y+z=1[\/latex] has intercepts\u00a0[latex](1, 0, 0)[\/latex], [latex](0, 1, 0)[\/latex], and [latex](0, 0, 1)[\/latex]. The region [latex]E[\/latex] looks like<\/p>\n<p style=\"text-align: center;\">[latex]{E} = {\\left \\{ {(x,y,z)}{\\mid}{0} \\ {\\leq} \\ {x} \\ {\\leq} \\ {1,0} \\ {\\leq} \\ {y} \\ {\\leq} \\ {{1-x},{0}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {1-x-y} \\right \\}}[\/latex].<\/p>\n<p id=\"fs-id1167793630666\">Hence the triple integral of the temperature is<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = {\\displaystyle\\int^{x=1}_{x=0}} \\ {\\displaystyle\\int^{y=1-x}_{y=0}} \\ {\\displaystyle\\int^{z=1-x-y}_{z=0}}(xy+8z+20){dz}{dy}{dx} = {\\frac{147}{40}}[\/latex].<\/p>\n<p>The volume evaluation is<\/p>\n<p style=\"text-align: center;\">[latex]{V}{(E)} = \\underset{E}{\\displaystyle\\iiint}{1dV} = {\\displaystyle\\int^{x=1}_{x=0}} \\ {\\displaystyle\\int^{y=1-x}_{y=0}} \\ {\\displaystyle\\int^{z=1-x-y}_{z=0}}{1dzdydx} = {\\frac{1}{6}}[\/latex].<\/p>\n<p id=\"fs-id1167794160233\">Hence the average value is [latex]{{T}_{\\text{ave}}} = {\\frac{147\/40}{1\/6}} = {\\frac{6(147)}{40}} = {\\frac{441}{20}}[\/latex] degrees Celsius.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p>Find the average value of the function [latex]f(x, y, z)=xyz[\/latex] over the cube with sides of length 4 units in the first octant with one vertex at the origin and edges parallel to the coordinate axes.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q151360489\">Show Solution<\/span><\/p>\n<div id=\"q151360489\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]f_{\\text{ave}}=8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>Watch the following video to see the worked solution to the above Try It<\/p>\n<div style=\"text-align: center;\"><iframe loading=\"lazy\" src=\"\/\/plugin.3playmedia.com\/show?mf=8197107&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=6U0zv9wquVo&amp;video_target=tpm-plugin-7r1ilu60-6U0zv9wquVo\" width=\"800px\" height=\"450px\" frameborder=\"0\" marginwidth=\"0px\" marginheight=\"0px\"><\/iframe><\/div>\n<div style=\"text-align: center;\">You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Calculus+3\/Calc+3+transcripts\/CP5.26_transcript.html\">transcript for \u201cCP 5.26\u201d here (opens in new window).<\/a><\/div>\n<\/div>\n<section data-depth=\"1\"><\/section>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1068\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>CP 5.26. <strong>Authored by<\/strong>: Ryan Melton. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":428269,"menu_order":18,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Calculus Volume 3\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\"},{\"type\":\"original\",\"description\":\"CP 5.26\",\"author\":\"Ryan Melton\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1068","chapter","type-chapter","status-publish","hentry"],"part":23,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1068","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/users\/428269"}],"version-history":[{"count":104,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1068\/revisions"}],"predecessor-version":[{"id":6357,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1068\/revisions\/6357"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/parts\/23"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1068\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1068"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1068"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1068"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1068"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}