Q. 10 Let f(x,聽y,z)聽be a continuous ... [FREE SOLUTION]

Chapter 13: Q. 10 (page 1054)

Let $f (x, y, z)$ be a continuous function of three variables, let role="math" localid="1650355225242" $惟_{x z} = {(x, z) | a 鈮� x 鈮� b a n d h_{1} (x) 鈮� z 鈮� h_{2} (x)}$ be a set of points in the $x z$ -plane, and let $惟 = {(x, y, z) | (x, z) 鈭� 惟_{x z} a n d g_{1} (x, z) 鈮� y 鈮� g_{2} (x, z)}$ be a set of points in 3-space. Find an iterated triple integral equal to the triple integral $\underset{惟}{鈭�} f (x, y, z) d V$ . How would your answer change if $惟_{x z} = {(x, z) | a 鈮� z 鈮� b a n d h_{1} (z) 鈮� x 鈮� h_{2} (z)}$ ?

Short Answer

Expert verified

If in $x z$ -plane $惟_{x z} = {(x, z) | a 鈮� x 鈮� b a n d h_{1} (x) 鈮� z 鈮� h_{2} (x)}$ ,then the triple integral becomes,

role="math" localid="1650355536570" $\underset{惟}{鈭�} f (x, y, z) d v = {鈭�}_{a}^{b} {鈭�}_{h_{1} (z)}^{h_{2} (z)} {鈭�}_{g_{1} (x, z)}^{g_{2} (x, z)} f (x, y, z) d y d x d z$ .

Step by step solution

Step 1. Given information

$惟_{x z} = {(x, z) | a 鈮� x 鈮� b a n d h_{1} (x) 鈮� z 鈮� h_{2} (x)}$ .

Step 2. Find an iterated triple integral which is equal to ∭Ω fx,y,zdV:

If in $x z$ -plane $惟_{x z} = {(x, z) | a 鈮� x 鈮� b a n d h_{1} (x) 鈮� z 鈮� h_{2} (x)}$ , then the triple integral becomes,

$\underset{惟}{鈭�} f (x, y, z) d v = {鈭�}_{a}^{b} {鈭�}_{h_{1} (z)}^{h_{2} (z)} {鈭�}_{g_{1} (x, z)}^{g_{2} (x, z)} f (x, y, z) d y d x d z$ .

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Find an iterated triple integral which is equal to ∭Ω fx,y,zdV:

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