{"id":1247,"date":"2025-10-15T15:50:36","date_gmt":"2025-10-15T15:50:36","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=1247"},"modified":"2025-10-31T15:22:58","modified_gmt":"2025-10-31T15:22:58","slug":"triple-integrals-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/triple-integrals-learn-it-2\/","title":{"raw":"Triple Integrals: Learn It 2","rendered":"Triple Integrals: Learn It 2"},"content":{"raw":"<h2 data-type=\"title\">Triple Integrals over a General Bounded Region<\/h2>\r\n<p class=\"whitespace-normal break-words\">We now expand the definition of the triple integral to compute a triple integral over a more general bounded region [latex]E[\/latex] in [latex]{\\mathbb{R}^3}[\/latex]. The general bounded regions we will consider are of three types.<\/p>\r\n<p class=\"whitespace-normal break-words\">First, let [latex]D[\/latex] be the bounded region that is a projection of [latex]E[\/latex] onto the [latex]xy[\/latex]-plane. Suppose the region [latex]E[\/latex] in [latex]{\\mathbb{R}^3}[\/latex] has the form:<\/p>\r\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]E = {{(x,y,z)}{\\mid}{(x,y)} \\ {\\in} \\ {{D},{u_1}{(x,y)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,y)} }[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">For two functions [latex]z=u_1(x, y)[\/latex] and [latex]z=u_2(x,y)[\/latex], such that [latex]{u_1}{(x,y)} \\ {\\leq} \\ {u_2}{(x,y)}[\/latex] for all [latex](x, y)[\/latex] in [latex]D[\/latex] as shown in the following figure.<\/p>\r\n\r\n[caption id=\"attachment_1375\" align=\"aligncenter\" width=\"429\"]<img class=\"size-full wp-image-1375\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144346\/5-4-3.jpeg\" alt=\"In x y z space, there is a shape E with top surface z = u2(x, y) and bottom surface z = u1(x, y). The bottom projects onto the x y plane as region D.\" width=\"429\" height=\"348\" \/> Figure 3. We can describe region [latex]E[\/latex] as the space between\u00a0[latex]u_{1}(x,y)[\/latex] and\u00a0[latex]u_{2}(x,y)[\/latex] above the projection\u00a0[latex]D[\/latex] of\u00a0[latex]E[\/latex] onto the\u00a0[latex]xy[\/latex]-plane.[\/caption]<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>triple integral over a general region<\/h3>\r\n<p id=\"fs-id1167793632073\">The triple integral of a continuous function [latex]f(x, y, z)[\/latex] over a general three-dimensional region:<\/p>\r\n<p style=\"text-align: center;\">[latex]E = \\{{(x,y,z)}{\\mid}{(x,y)} \\ {\\in} \\ {{D},{u_1}{(x,y)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,y)} \\}[\/latex]<\/p>\r\n<span style=\"font-size: 1rem; text-align: initial;\">in [latex]{{\\mathbb{R}}^3}[\/latex],\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">where [latex]D[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is the projection of [latex]E[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">onto the [latex]xy[\/latex]-<\/span><span style=\"font-size: 1rem; text-align: initial;\">plane, is:<\/span>\r\n<p style=\"text-align: center;\">[latex]{\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = \\underset{D}{\\displaystyle\\iint}{[{\\displaystyle\\int^{{u_2}{(x,y)}}_{{u_1}{(x,y)}}}{f}{(x,y,z)}{dz} ]}{dA}}[\/latex]<\/p>\r\n\r\n<\/section>\r\n<p id=\"fs-id1167793895376\">Similarly, we can consider a general bounded region [latex]D[\/latex] in the [latex]xy[\/latex]-plane and two functions [latex]{y} = {u_1}{(x,z)}[\/latex] and [latex]{y} = {u_2}{(x,z)}[\/latex] such that [latex]{u_1}{(x,z)} \\ {\\leq} \\ {u_2}{(x,z)}[\/latex] for all [latex](x, z)[\/latex] in [latex]D[\/latex]. Then we can describe the solid region [latex]E[\/latex] in [latex]{{\\mathbb{R}}^3}[\/latex] as:<\/p>\r\n<p style=\"text-align: center;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{(x,z)} \\ {\\in} \\ {{D},{u_1}{(x,z)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,z)} \\right \\}[\/latex]<\/p>\r\nwhere [latex]D[\/latex] is the projection of [latex]E[\/latex] onto the [latex]xy[\/latex]-plane and the triple integral is:\r\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = \\underset{D}{\\displaystyle\\iint}{\\left [{\\displaystyle\\int^{{u_2}{(x,z)}}_{{u_1}{(x,z)}}}{f}{(x,y,z)}{dy} \\right ]}{dA}[\/latex].<\/p>\r\n<span style=\"font-size: 1rem; text-align: initial;\">Finally, if [latex]D[\/latex] is a general bounded region in the [latex]yz[\/latex]-plane and we have two functions [latex]{x} = {u_1}{(y,z)}[\/latex] and [latex]{x} = {u_2}{(y,z)}[\/latex] such that [latex]{u_1}{(y,z)} \\ {\\leq} \\ {u_2}{(y,z)}[\/latex] for all [latex](y, z)[\/latex] in [latex]D[\/latex], then the solid region [latex]E[\/latex] in [latex]{{\\mathbb{R}}^3}[\/latex] can be described as:<\/span>\r\n<p style=\"text-align: center;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{(y,z)} \\ {\\in} \\ {{D},{u_1}{(y,z)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(y,z)} \\right \\}[\/latex]<\/p>\r\nwhere [latex]D[\/latex] is the projection of [latex]E[\/latex] onto the [latex]yz[\/latex]-plane and the triple integral is:\r\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = \\underset{D}{\\displaystyle\\iint}{\\left [{\\displaystyle\\int^{{u_2}{(y,z)}}_{{u_1}{(y,z)}}}{f}{(x,y,z)}{dx} \\right ]}{dA}[\/latex].<\/p>\r\n\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Setting Up Triple Integrals with Type I and Type II Regions<\/h3>\r\n<span style=\"font-size: 1rem; text-align: initial;\">Note that the region [latex]D[\/latex] in any of the planes may be of Type I or Type II<\/span><span style=\"font-size: 1rem; text-align: initial;\">. <\/span>\r\n\r\n<section class=\"textbox recall\" aria-label=\"Recall\"><strong>Type I and Type II Regions<\/strong>\r\n<p class=\"whitespace-normal break-words\">A <strong>Type I region<\/strong> in the [latex]xy[\/latex]-plane has the form: [latex]a \\leq x \\leq b[\/latex] and [latex]g_1(x) \\leq y \\leq g_2(x)[\/latex]. The bounds on [latex]y[\/latex] depend on [latex]x[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">A <strong>Type II region<\/strong> in the [latex]xy[\/latex]-plane has the form: [latex]c \\leq y \\leq d[\/latex] and [latex]h_1(y) \\leq x \\leq h_2(y)[\/latex]. The bounds on [latex]x[\/latex] depend on [latex]y[\/latex].<\/p>\r\n<p class=\"whitespace-normal break-words\">These same classifications apply to regions in the [latex]xz[\/latex]-plane and [latex]yz[\/latex]-plane.<\/p>\r\n\r\n<\/section><span style=\"font-size: 1rem; text-align: initial;\">If [latex]D[\/latex] in the [latex]xy[\/latex]-plane is of <strong>Type I<\/strong> (<\/span>Figure 4<span style=\"font-size: 1rem; text-align: initial;\">), then:<\/span>\r\n<p style=\"text-align: center;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{a} \\ {\\leq} \\ {x} \\ {\\leq} \\ {b,{g_1}{(x)}} \\ {\\leq} \\ {y} \\ {\\leq} \\ {g_2{(x)}},{{u_1}{(x,y)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,y)} \\right \\}[\/latex].<\/p>\r\n\r\n<section data-depth=\"1\"><\/section><section data-depth=\"1\">[caption id=\"attachment_1376\" align=\"aligncenter\" width=\"571\"]<img class=\"wp-image-1376 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144428\/5-4-4.jpeg\" alt=\"In x y z space, there is a complex shape E with top surface z = u2(x, y) and bottom surface z = u1(x, y). The bottom projects onto the xy plane as region D with boundaries x = a, x = b, y = g1(x), and y = g2(x).\" width=\"571\" height=\"443\" \/> Figure 4. A box\u00a0[latex]E[\/latex] where the projection\u00a0[latex]D[\/latex] in the\u00a0[latex]xy[\/latex]-plane is of Type I.[\/caption]Then the triple integral becomes:\r\n<p style=\"text-align: center;\"><span style=\"font-size: 1em;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = {\\displaystyle\\int_{a}^{b}} \\ {\\displaystyle\\int_{{g_1}{(x)}}^{{g_2}{(x)}}} \\ {\\displaystyle\\int_{{u_1}{(x,y)}}^{{u_2}{(x,y)}}}{f}{(x,y,z)}{dz}{dy}{dx}[\/latex].<\/span><\/p>\r\n<p style=\"text-align: left;\"><span style=\"font-size: 1em;\">If [latex]D[\/latex] in the [latex]xy[\/latex]-plane is of <strong>Type II<\/strong> (Figure 5), then:<\/span><\/p>\r\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem; text-align: initial;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{c} \\ {\\leq} \\ {x} \\ {\\leq} \\ {d,{h_1}{(x)}} \\ {\\leq} \\ {y} \\ {\\leq} \\ {h_2{(x)}},{{u_1}{(x,y)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,y)} \\right \\}[\/latex].<\/span><\/p>\r\n\r\n<\/section>[caption id=\"attachment_1377\" align=\"aligncenter\" width=\"444\"]<img class=\"size-full wp-image-1377\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144556\/5-4-5.jpeg\" alt=\"In x y z space, there is a complex shape E with top surface z = u2(x, y) and bottom surface z = u1(x, y). The bottom projects onto the xy plane as region D with boundaries y = c, y = d, x = h1(y), and x = h2(y).\" width=\"444\" height=\"397\" \/> Figure 5.\u00a0A box\u00a0[latex]E[\/latex] where the projection\u00a0[latex]D[\/latex] in the\u00a0[latex]xy[\/latex]-plane is of Type II.[\/caption]Then the triple integral becomes:\r\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = {\\displaystyle\\int^{y=d}_{y=c}}{\\displaystyle\\int^{x={{h_2}{(y)}}}_{x={h_1}{(y)}}}{\\displaystyle\\int^{z={{u_2}{(x,y)}}}_{z={u_1}{(x,y)}}}{f}{(x,y,z)}{dz}{dx}{dy}[\/latex].<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the triple integral of the function [latex]f(x, y, z)=5x-3y[\/latex] over the solid tetrahedron bounded by the planes\u00a0[latex]x=0, \\ y=0, \\ z=0,[\/latex] and\u00a0[latex]x+y+z=1[\/latex].[reveal-answer q=\"331457700\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"331457700\"]Figure 6\u00a0shows the solid tetrahedron [latex]E[\/latex] and its projection [latex]D[\/latex] on the [latex]xy[\/latex]-plane.[caption id=\"attachment_1378\" align=\"aligncenter\" width=\"685\"]<img class=\"size-full wp-image-1378\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144633\/5-4-6.jpeg\" alt=\"In x y z space, there is a solid E with boundaries being the x y, z y, and x z planes and z = 1 minus x minus y. The points are the origin, (1, 0, 0), (0, 0, 1), and (0, 1, 0). Its surface on the x y plane is shown as being a rectangle marked D with line y = 1 minus x. Additionally, there is a vertical line shown on D.\" width=\"685\" height=\"357\" \/> Figure 6.\u00a0The solid [latex]E[\/latex] has a projection\u00a0[latex]D[\/latex] on the\u00a0[latex]xy[\/latex]-plane of Type I.[\/caption]\r\n<p id=\"fs-id1167794051980\">We can describe the solid region tetrahedron as<\/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-id1167793887543\">Hence, the triple integral 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}}{(5x-3y)}{dz}{dy}{dx}[\/latex].<\/p>\r\n<p id=\"fs-id1167793423320\">To simplify the calculation, first evaluate the integral [latex]{\\displaystyle\\int^{z=1-x-y}_{z=0}}{(5x-3y)}{dz}[\/latex]. We have<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\displaystyle\\int^{z=1-x-y}_{z=0}}{(5x-3y)}{dz} = {(5x-3y)}{(1-x-y)}[\/latex].<\/p>\r\n<p id=\"fs-id1167793421147\">Now evaluate the integral [latex]{\\displaystyle\\int^{y=1-x}_{y=0}}{(5x-3y)}{(1-x-y)}{dy}[\/latex], obtaining<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\displaystyle\\int^{y=1-x}_{y=0}}{(5x-3y)}{(1-x-y)}{dy} = {\\frac{1}{2}}{(x-1)^2}{(6x-1)}[\/latex].<\/p>\r\n<p id=\"fs-id1167794045783\">Finally, evaluate<\/p>\r\n<p style=\"text-align: center;\">[latex]{\\displaystyle\\int^{x=1}_{x=0}}{\\frac{1}{2}}{(x-1)^2}{(6x-1)}{dx} = {\\frac{1}{12}}[\/latex].<\/p>\r\n<span style=\"font-size: 1rem; text-align: initial;\">Putting it all together, we have<\/span>\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}}{(5x-3y)}{dz}{dy}{dx} = {\\frac{1}{12}}[\/latex].<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Using Triple Integrals to Find Volume<\/h3>\r\n<p class=\"whitespace-normal break-words\">Just as we used the double integral [latex]\\underset{D}{\\displaystyle\\iint}{1}{dA}[\/latex] to find the area of a general bounded region [latex]D[\/latex], we can use [latex]\\underset{E}{\\displaystyle\\iiint}{1}{dV}[\/latex] to find the volume of a general solid bounded region [latex]E[\/latex]. The next example illustrates the method.<\/p>\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\">\r\n<p id=\"fs-id1167793504416\">Find the volume of a right pyramid that has the square base in the [latex]xy[\/latex]-plane [latex][-1,1] \\ {\\times} \\ [-1,1][\/latex] and vertex at the point [latex](0, 0, 1)[\/latex] as shown in the following figure.<\/p>\r\n\r\n\r\n[caption id=\"attachment_1379\" align=\"aligncenter\" width=\"543\"]<img class=\"size-full wp-image-1379\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144721\/5-4-7.jpeg\" alt=\"In x y z space, there is a pyramid with a square base centered at the origin. The apex of the pyramid is (0, 0, 1).\" width=\"543\" height=\"527\" \/> Figure 7.\u00a0Finding the volume of a pyramid with a square base.[\/caption]\r\n\r\n[reveal-answer q=\"425689014\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"425689014\"]\r\n<p id=\"fs-id1167794005598\">In this pyramid the value of [latex]z[\/latex] changes from 0 to 1 and at each height [latex]z[\/latex], the cross section of the pyramid for any value of [latex]z[\/latex] is the square [latex][-1+z,1-z] \\ {\\times} \\ [-1+z,1-z][\/latex]. Hence, the volume of the pyramid is [latex]\\underset{E}{\\displaystyle\\iiint}{1}{dV}[\/latex] where<\/p>\r\n<p style=\"text-align: center;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{0} \\ {\\leq} \\ {z} \\ {\\leq} \\ {1,-1+z} \\ {\\leq} \\ {y} \\ {\\leq} \\ {1-z,-1+z} \\ {\\leq} \\ {x} \\ {\\leq} \\ {1-z} \\right \\}[\/latex].<\/p>\r\n<p id=\"fs-id1167793442473\">Thus, we have<\/p>\r\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{1}{dV} = {\\displaystyle\\int^{z=1}_{z=0}}{\\displaystyle\\int^{y={1-z}}_{y=-1+z}}{\\displaystyle\\int^{x={1-z}}_{x=1+z}}{1}{dx}{dy}{dz} = {\\displaystyle\\int^{z=1}_{z=0}}{\\displaystyle\\int^{y={1-z}}_{y=-1+z}}{(2-2z)}{dy}{dz} = {\\displaystyle\\int^{z=1}_{z=0}}{(2-2z)^2}{dz} = {\\frac{4}{3}}[\/latex].<\/p>\r\n<p id=\"fs-id1167793637795\">Hence, the volume of the pyramid is\u00a0[latex]{\\frac{4}{3}}[\/latex] cubic units.<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"<h2 data-type=\"title\">Triple Integrals over a General Bounded Region<\/h2>\n<p class=\"whitespace-normal break-words\">We now expand the definition of the triple integral to compute a triple integral over a more general bounded region [latex]E[\/latex] in [latex]{\\mathbb{R}^3}[\/latex]. The general bounded regions we will consider are of three types.<\/p>\n<p class=\"whitespace-normal break-words\">First, let [latex]D[\/latex] be the bounded region that is a projection of [latex]E[\/latex] onto the [latex]xy[\/latex]-plane. Suppose the region [latex]E[\/latex] in [latex]{\\mathbb{R}^3}[\/latex] has the form:<\/p>\n<p class=\"whitespace-normal break-words\" style=\"text-align: center;\">[latex]E = {{(x,y,z)}{\\mid}{(x,y)} \\ {\\in} \\ {{D},{u_1}{(x,y)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,y)} }[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">For two functions [latex]z=u_1(x, y)[\/latex] and [latex]z=u_2(x,y)[\/latex], such that [latex]{u_1}{(x,y)} \\ {\\leq} \\ {u_2}{(x,y)}[\/latex] for all [latex](x, y)[\/latex] in [latex]D[\/latex] as shown in the following figure.<\/p>\n<figure id=\"attachment_1375\" aria-describedby=\"caption-attachment-1375\" style=\"width: 429px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1375\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144346\/5-4-3.jpeg\" alt=\"In x y z space, there is a shape E with top surface z = u2(x, y) and bottom surface z = u1(x, y). The bottom projects onto the x y plane as region D.\" width=\"429\" height=\"348\" \/><figcaption id=\"caption-attachment-1375\" class=\"wp-caption-text\">Figure 3. We can describe region [latex]E[\/latex] as the space between\u00a0[latex]u_{1}(x,y)[\/latex] and\u00a0[latex]u_{2}(x,y)[\/latex] above the projection\u00a0[latex]D[\/latex] of\u00a0[latex]E[\/latex] onto the\u00a0[latex]xy[\/latex]-plane.<\/figcaption><\/figure>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>triple integral over a general region<\/h3>\n<p id=\"fs-id1167793632073\">The triple integral of a continuous function [latex]f(x, y, z)[\/latex] over a general three-dimensional region:<\/p>\n<p style=\"text-align: center;\">[latex]E = \\{{(x,y,z)}{\\mid}{(x,y)} \\ {\\in} \\ {{D},{u_1}{(x,y)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,y)} \\}[\/latex]<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">in [latex]{{\\mathbb{R}}^3}[\/latex],\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">where [latex]D[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">is the projection of [latex]E[\/latex]\u00a0<\/span><span style=\"font-size: 1rem; text-align: initial;\">onto the [latex]xy[\/latex]&#8211;<\/span><span style=\"font-size: 1rem; text-align: initial;\">plane, is:<\/span><\/p>\n<p style=\"text-align: center;\">[latex]{\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = \\underset{D}{\\displaystyle\\iint}{[{\\displaystyle\\int^{{u_2}{(x,y)}}_{{u_1}{(x,y)}}}{f}{(x,y,z)}{dz} ]}{dA}}[\/latex]<\/p>\n<\/section>\n<p id=\"fs-id1167793895376\">Similarly, we can consider a general bounded region [latex]D[\/latex] in the [latex]xy[\/latex]-plane and two functions [latex]{y} = {u_1}{(x,z)}[\/latex] and [latex]{y} = {u_2}{(x,z)}[\/latex] such that [latex]{u_1}{(x,z)} \\ {\\leq} \\ {u_2}{(x,z)}[\/latex] for all [latex](x, z)[\/latex] in [latex]D[\/latex]. Then we can describe the solid region [latex]E[\/latex] in [latex]{{\\mathbb{R}}^3}[\/latex] as:<\/p>\n<p style=\"text-align: center;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{(x,z)} \\ {\\in} \\ {{D},{u_1}{(x,z)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,z)} \\right \\}[\/latex]<\/p>\n<p>where [latex]D[\/latex] is the projection of [latex]E[\/latex] onto the [latex]xy[\/latex]-plane and the triple integral is:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = \\underset{D}{\\displaystyle\\iint}{\\left [{\\displaystyle\\int^{{u_2}{(x,z)}}_{{u_1}{(x,z)}}}{f}{(x,y,z)}{dy} \\right ]}{dA}[\/latex].<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Finally, if [latex]D[\/latex] is a general bounded region in the [latex]yz[\/latex]-plane and we have two functions [latex]{x} = {u_1}{(y,z)}[\/latex] and [latex]{x} = {u_2}{(y,z)}[\/latex] such that [latex]{u_1}{(y,z)} \\ {\\leq} \\ {u_2}{(y,z)}[\/latex] for all [latex](y, z)[\/latex] in [latex]D[\/latex], then the solid region [latex]E[\/latex] in [latex]{{\\mathbb{R}}^3}[\/latex] can be described as:<\/span><\/p>\n<p style=\"text-align: center;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{(y,z)} \\ {\\in} \\ {{D},{u_1}{(y,z)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(y,z)} \\right \\}[\/latex]<\/p>\n<p>where [latex]D[\/latex] is the projection of [latex]E[\/latex] onto the [latex]yz[\/latex]-plane and the triple integral is:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = \\underset{D}{\\displaystyle\\iint}{\\left [{\\displaystyle\\int^{{u_2}{(y,z)}}_{{u_1}{(y,z)}}}{f}{(x,y,z)}{dx} \\right ]}{dA}[\/latex].<\/p>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Setting Up Triple Integrals with Type I and Type II Regions<\/h3>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Note that the region [latex]D[\/latex] in any of the planes may be of Type I or Type II<\/span><span style=\"font-size: 1rem; text-align: initial;\">. <\/span><\/p>\n<section class=\"textbox recall\" aria-label=\"Recall\"><strong>Type I and Type II Regions<\/strong><\/p>\n<p class=\"whitespace-normal break-words\">A <strong>Type I region<\/strong> in the [latex]xy[\/latex]-plane has the form: [latex]a \\leq x \\leq b[\/latex] and [latex]g_1(x) \\leq y \\leq g_2(x)[\/latex]. The bounds on [latex]y[\/latex] depend on [latex]x[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">A <strong>Type II region<\/strong> in the [latex]xy[\/latex]-plane has the form: [latex]c \\leq y \\leq d[\/latex] and [latex]h_1(y) \\leq x \\leq h_2(y)[\/latex]. The bounds on [latex]x[\/latex] depend on [latex]y[\/latex].<\/p>\n<p class=\"whitespace-normal break-words\">These same classifications apply to regions in the [latex]xz[\/latex]-plane and [latex]yz[\/latex]-plane.<\/p>\n<\/section>\n<p><span style=\"font-size: 1rem; text-align: initial;\">If [latex]D[\/latex] in the [latex]xy[\/latex]-plane is of <strong>Type I<\/strong> (<\/span>Figure 4<span style=\"font-size: 1rem; text-align: initial;\">), then:<\/span><\/p>\n<p style=\"text-align: center;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{a} \\ {\\leq} \\ {x} \\ {\\leq} \\ {b,{g_1}{(x)}} \\ {\\leq} \\ {y} \\ {\\leq} \\ {g_2{(x)}},{{u_1}{(x,y)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,y)} \\right \\}[\/latex].<\/p>\n<section data-depth=\"1\"><\/section>\n<section data-depth=\"1\">\n<figure id=\"attachment_1376\" aria-describedby=\"caption-attachment-1376\" style=\"width: 571px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-1376 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144428\/5-4-4.jpeg\" alt=\"In x y z space, there is a complex shape E with top surface z = u2(x, y) and bottom surface z = u1(x, y). The bottom projects onto the xy plane as region D with boundaries x = a, x = b, y = g1(x), and y = g2(x).\" width=\"571\" height=\"443\" \/><figcaption id=\"caption-attachment-1376\" class=\"wp-caption-text\">Figure 4. A box\u00a0[latex]E[\/latex] where the projection\u00a0[latex]D[\/latex] in the\u00a0[latex]xy[\/latex]-plane is of Type I.<\/figcaption><\/figure>\n<p>Then the triple integral becomes:<\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 1em;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = {\\displaystyle\\int_{a}^{b}} \\ {\\displaystyle\\int_{{g_1}{(x)}}^{{g_2}{(x)}}} \\ {\\displaystyle\\int_{{u_1}{(x,y)}}^{{u_2}{(x,y)}}}{f}{(x,y,z)}{dz}{dy}{dx}[\/latex].<\/span><\/p>\n<p style=\"text-align: left;\"><span style=\"font-size: 1em;\">If [latex]D[\/latex] in the [latex]xy[\/latex]-plane is of <strong>Type II<\/strong> (Figure 5), then:<\/span><\/p>\n<p style=\"text-align: center;\"><span style=\"font-size: 1rem; text-align: initial;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{c} \\ {\\leq} \\ {x} \\ {\\leq} \\ {d,{h_1}{(x)}} \\ {\\leq} \\ {y} \\ {\\leq} \\ {h_2{(x)}},{{u_1}{(x,y)}} \\ {\\leq} \\ {z} \\ {\\leq} \\ {u_2}{(x,y)} \\right \\}[\/latex].<\/span><\/p>\n<\/section>\n<figure id=\"attachment_1377\" aria-describedby=\"caption-attachment-1377\" style=\"width: 444px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1377\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144556\/5-4-5.jpeg\" alt=\"In x y z space, there is a complex shape E with top surface z = u2(x, y) and bottom surface z = u1(x, y). The bottom projects onto the xy plane as region D with boundaries y = c, y = d, x = h1(y), and x = h2(y).\" width=\"444\" height=\"397\" \/><figcaption id=\"caption-attachment-1377\" class=\"wp-caption-text\">Figure 5.\u00a0A box\u00a0[latex]E[\/latex] where the projection\u00a0[latex]D[\/latex] in the\u00a0[latex]xy[\/latex]-plane is of Type II.<\/figcaption><\/figure>\n<p>Then the triple integral becomes:<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV} = {\\displaystyle\\int^{y=d}_{y=c}}{\\displaystyle\\int^{x={{h_2}{(y)}}}_{x={h_1}{(y)}}}{\\displaystyle\\int^{z={{u_2}{(x,y)}}}_{z={u_1}{(x,y)}}}{f}{(x,y,z)}{dz}{dx}{dy}[\/latex].<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">Evaluate the triple integral of the function [latex]f(x, y, z)=5x-3y[\/latex] over the solid tetrahedron bounded by the planes\u00a0[latex]x=0, \\ y=0, \\ z=0,[\/latex] and\u00a0[latex]x+y+z=1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q331457700\">Show Solution<\/button><\/p>\n<div id=\"q331457700\" class=\"hidden-answer\" style=\"display: none\">Figure 6\u00a0shows the solid tetrahedron [latex]E[\/latex] and its projection [latex]D[\/latex] on the [latex]xy[\/latex]-plane.<\/p>\n<figure id=\"attachment_1378\" aria-describedby=\"caption-attachment-1378\" style=\"width: 685px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1378\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144633\/5-4-6.jpeg\" alt=\"In x y z space, there is a solid E with boundaries being the x y, z y, and x z planes and z = 1 minus x minus y. The points are the origin, (1, 0, 0), (0, 0, 1), and (0, 1, 0). Its surface on the x y plane is shown as being a rectangle marked D with line y = 1 minus x. Additionally, there is a vertical line shown on D.\" width=\"685\" height=\"357\" \/><figcaption id=\"caption-attachment-1378\" class=\"wp-caption-text\">Figure 6.\u00a0The solid [latex]E[\/latex] has a projection\u00a0[latex]D[\/latex] on the\u00a0[latex]xy[\/latex]-plane of Type I.<\/figcaption><\/figure>\n<p id=\"fs-id1167794051980\">We can describe the solid region tetrahedron as<\/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-id1167793887543\">Hence, the triple integral 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}}{(5x-3y)}{dz}{dy}{dx}[\/latex].<\/p>\n<p id=\"fs-id1167793423320\">To simplify the calculation, first evaluate the integral [latex]{\\displaystyle\\int^{z=1-x-y}_{z=0}}{(5x-3y)}{dz}[\/latex]. We have<\/p>\n<p style=\"text-align: center;\">[latex]{\\displaystyle\\int^{z=1-x-y}_{z=0}}{(5x-3y)}{dz} = {(5x-3y)}{(1-x-y)}[\/latex].<\/p>\n<p id=\"fs-id1167793421147\">Now evaluate the integral [latex]{\\displaystyle\\int^{y=1-x}_{y=0}}{(5x-3y)}{(1-x-y)}{dy}[\/latex], obtaining<\/p>\n<p style=\"text-align: center;\">[latex]{\\displaystyle\\int^{y=1-x}_{y=0}}{(5x-3y)}{(1-x-y)}{dy} = {\\frac{1}{2}}{(x-1)^2}{(6x-1)}[\/latex].<\/p>\n<p id=\"fs-id1167794045783\">Finally, evaluate<\/p>\n<p style=\"text-align: center;\">[latex]{\\displaystyle\\int^{x=1}_{x=0}}{\\frac{1}{2}}{(x-1)^2}{(6x-1)}{dx} = {\\frac{1}{12}}[\/latex].<\/p>\n<p><span style=\"font-size: 1rem; text-align: initial;\">Putting it all together, we have<\/span><\/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}}{(5x-3y)}{dz}{dy}{dx} = {\\frac{1}{12}}[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/section>\n<h3 class=\"text-lg font-bold text-text-100 mt-1 -mb-1.5\">Using Triple Integrals to Find Volume<\/h3>\n<p class=\"whitespace-normal break-words\">Just as we used the double integral [latex]\\underset{D}{\\displaystyle\\iint}{1}{dA}[\/latex] to find the area of a general bounded region [latex]D[\/latex], we can use [latex]\\underset{E}{\\displaystyle\\iiint}{1}{dV}[\/latex] to find the volume of a general solid bounded region [latex]E[\/latex]. The next example illustrates the method.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\">\n<p id=\"fs-id1167793504416\">Find the volume of a right pyramid that has the square base in the [latex]xy[\/latex]-plane [latex][-1,1] \\ {\\times} \\ [-1,1][\/latex] and vertex at the point [latex](0, 0, 1)[\/latex] as shown in the following figure.<\/p>\n<figure id=\"attachment_1379\" aria-describedby=\"caption-attachment-1379\" style=\"width: 543px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-1379\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5667\/2021\/11\/25144721\/5-4-7.jpeg\" alt=\"In x y z space, there is a pyramid with a square base centered at the origin. The apex of the pyramid is (0, 0, 1).\" width=\"543\" height=\"527\" \/><figcaption id=\"caption-attachment-1379\" class=\"wp-caption-text\">Figure 7.\u00a0Finding the volume of a pyramid with a square base.<\/figcaption><\/figure>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q425689014\">Show Solution<\/button><\/p>\n<div id=\"q425689014\" class=\"hidden-answer\" style=\"display: none\">\n<p id=\"fs-id1167794005598\">In this pyramid the value of [latex]z[\/latex] changes from 0 to 1 and at each height [latex]z[\/latex], the cross section of the pyramid for any value of [latex]z[\/latex] is the square [latex][-1+z,1-z] \\ {\\times} \\ [-1+z,1-z][\/latex]. Hence, the volume of the pyramid is [latex]\\underset{E}{\\displaystyle\\iiint}{1}{dV}[\/latex] where<\/p>\n<p style=\"text-align: center;\">[latex]{E} = \\left \\{{(x,y,z)}{\\mid}{0} \\ {\\leq} \\ {z} \\ {\\leq} \\ {1,-1+z} \\ {\\leq} \\ {y} \\ {\\leq} \\ {1-z,-1+z} \\ {\\leq} \\ {x} \\ {\\leq} \\ {1-z} \\right \\}[\/latex].<\/p>\n<p id=\"fs-id1167793442473\">Thus, we have<\/p>\n<p style=\"text-align: center;\">[latex]\\underset{E}{\\displaystyle\\iiint}{1}{dV} = {\\displaystyle\\int^{z=1}_{z=0}}{\\displaystyle\\int^{y={1-z}}_{y=-1+z}}{\\displaystyle\\int^{x={1-z}}_{x=1+z}}{1}{dx}{dy}{dz} = {\\displaystyle\\int^{z=1}_{z=0}}{\\displaystyle\\int^{y={1-z}}_{y=-1+z}}{(2-2z)}{dy}{dz} = {\\displaystyle\\int^{z=1}_{z=0}}{(2-2z)^2}{dz} = {\\frac{4}{3}}[\/latex].<\/p>\n<p id=\"fs-id1167793637795\">Hence, the volume of the pyramid is\u00a0[latex]{\\frac{4}{3}}[\/latex] cubic units.<\/p>\n<\/div>\n<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":6,"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-1247","chapter","type-chapter","status-publish","hentry"],"part":434,"module-header":"- Select Header -","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\/1247","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":9,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1247\/revisions"}],"predecessor-version":[{"id":1909,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapters\/1247\/revisions\/1909"}],"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\/1247\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/media?parent=1247"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/pressbooks\/v2\/chapter-type?post=1247"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/contributor?post=1247"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus3\/wp-json\/wp\/v2\/license?post=1247"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}