{"id":1249,"date":"2025-10-15T15:50:40","date_gmt":"2025-10-15T15:50:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1249"},"modified":"2025-10-31T15:26:23","modified_gmt":"2025-10-31T15:26:23","slug":"triple-integrals-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/triple-integrals-learn-it-4\/","title":{"raw":"Triple Integrals: Learn It 4","rendered":"Triple Integrals: Learn It 4"},"content":{"raw":"

Average Value of a Function of Three Variables<\/h2>\r\nRecall 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.\r\n\r\n
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average value of a function of three variables<\/h3>\r\n

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

[latex]{f_{\\text{ave}}} = {\\frac{1}{V(E)}}\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV}[\/latex].<\/p>\r\n

Note that the volume is <\/em>[latex]{V(E)} = \\underset{E}{\\displaystyle\\iiint}{1}{dV}[\/latex].<\/p>\r\n\r\n<\/section>

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.[reveal-answer q=\"067483257\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"067483257\"]\r\n

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

[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

Hence the triple integral of the temperature is<\/p>\r\n

[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

[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

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<\/section>","rendered":"

Average Value of a Function of Three Variables<\/h2>\n

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

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average value of a function of three variables<\/h3>\n

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

[latex]{f_{\\text{ave}}} = {\\frac{1}{V(E)}}\\underset{E}{\\displaystyle\\iiint}{f}{(x,y,z)}{dV}[\/latex].<\/p>\n

Note that the volume is <\/em>[latex]{V(E)} = \\underset{E}{\\displaystyle\\iiint}{1}{dV}[\/latex].<\/p>\n<\/section>\n

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