Q. 8 Let f(x,聽y,z)聽be a continuous ... [FREE SOLUTION]

Chapter 13: Q. 8 (page 1054)

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

Short Answer

Expert verified

If $惟_{x y} = {(x, y) | a 鈮� y 鈮� b a n d h_{1} (y) 鈮� x 鈮� h_{2} (y)}$ then the triple integral becomes,role="math" localid="1650351304797" $\underset{惟}{鈭� f} (x, y, z) d v = {鈭�}_{a}^{b} {鈭�}_{h_{1} (y)}^{h_{2} (y)} {鈭�}_{g_{1} (x, y)}^{g_{2} (x, y)} d z d x d y$ .

Step by step solution

Step 1. Given information

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

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

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

Since $a 鈮� y 鈮� b$ and $h_{1} (y) 鈮� x 鈮� h_{2} (y)$ and $g_{1} (x, y) 鈮� z 鈮� g_{2} (x, y)$ .

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Short Answer

Step by step solution

Step 1. Given information

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

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