Q. 9 Let f(x,聽y,z)聽be a continuous ... [FREE SOLUTION]

Chapter 13: Q. 9 (page 1054)

Let $f (x, y, z)$ be a continuous function of three variables, let localid="1650352548375" $惟_{y z} = {(y, z) | a 鈮� y 鈮� b a n d h_{1} (y) 鈮� z 鈮� h_{2} (y)}$ be a set of points in the $y z$ -plane, and let localid="1650354983053" $惟 = {(x, y, z) | (y, z) 鈭� 惟_{y z} a n d g_{1} (y, z) 鈮� x 鈮� g_{2} (y, z)}$ be a set of points in $3$ -space. Find an iterated triple integral equal to the triple integral localid="1650353288865" $\underset{惟}{鈭�} f (x, y, z) d V$ . How would your answer change iflocalid="1650352747263" $惟_{y z} = {(y, z) | a 鈮� z 鈮� b a n d h_{1} (z) 鈮� x 鈮� h_{2} (z)}$ ?

Short Answer

Expert verified

If in $y z$ -plane,localid="1650353321170" $惟_{y z} = {(y, z) | a 鈮� y 鈮� b a n d h_{1} (y) 鈮� z 鈮� h_{2} (y)}$ , then the triple integral becomes,
localid="1650355042503" $\underset{惟}{鈭�} f (x, y, z) d v = {鈭�}_{a}^{b} {鈭�}_{h_{1} (z)}^{h_{2} (z)} {鈭�}_{g_{1} (x, y)}^{g_{2} (x, y)} f (x, y, z) d x d y d z$ .

Because, localid="1650355058244" $惟 = {(x, y, z) | (y, z) 鈭� 惟_{y z} a n d g_{1} (y, z) 鈮� x 鈮� g_{2} (y, z)}$ .

Step by step solution

Step 1. Given information

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

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

Step 2. Find an iterated integral which is equal to ∭Ωf(x, y,z) dV:

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

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

Because,localid="1650355100648" $惟 = {(x, y, z) | (y, z) 鈭� 惟_{y z} a n d g_{1} (y, z) 鈮� x 鈮� g_{2} (y, z)}$ .

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Short Answer

Step by step solution

Step 1. Given information

Step 2. Find an iterated integral which is equal to ∭Ωf(x, y,z) dV:

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