Chapter 13: Q. 57 (page 1005)

Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = y^{3} e^{x y^{2}} W h e r e R = {(x, y) | 0 \leq x \leq 2, - 2 \leq y \leq 3}$

Short Answer

Expert verified

The volume is $16414245 c u b i c u n i t s$

Step by step solution

Given information

We are given an integral as

$f (x, y) = y^{3} e^{x y^{2}} W h e r e R = {(x, y) | 0 \leq x \leq 2, - 2 \leq y \leq 3}$

Evaluate the integral

We have

$f (x, y) > 0 w h e n R_{1} = {(x, y) | 0 \leq x \leq 2, 0 \leq y \leq 3} a n d f (x, y) < 0 w h e n R_{2} = {(x, y) | 0 \leq x \leq 2, - 2 \leq y \leq 0}$

Therefore

$\int_{R} y^{3} e^{x y^{2}} d A = \int_{R_{1}} y^{3} e^{x y^{2}} d A - \int_{R_{2}} y^{3} e^{x y^{2}} d A = \int_{0}^{3} \int_{0}^{2} y^{3} e^{x y^{2}} d x d y - \int_{- 2}^{0} \int_{0}^{2} y^{3} e^{x y^{2}} d x d y = \int_{0}^{3} {[y e^{x y^{2}}]}^{2}_{0} d y - \int_{- 2}^{0} {[y e^{x y^{2}}]}^{2}_{0} d y = \int_{0}^{3} (y e^{2 y^{2}} - y) d y - \int_{- 2}^{0} (y e^{2 y^{2}} - y) d y = 16414245 c u b i c u n i t s$

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Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = y^{3} e^{x y^{2}} W h e r e R = {(x, y) | 0 \leq x \leq 2, - 2 \leq y \leq 3}$

Short Answer

Step by step solution

Given information

Evaluate the integral

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Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-planef(x,y)=y3exy2WhereR={(x,y)|0≤x≤2,-2≤y≤3}

Short Answer

Step by step solution

Given information

Evaluate the integral

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Applied Mathematics

Decision Maths

Pure Maths

Statistics

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = y^{3} e^{x y^{2}} W h e r e R = {(x, y) | 0 \leq x \leq 2, - 2 \leq y \leq 3}$