Chapter 13: Q. 58 (page 1005)

Find the signed volume between the graph of the function and the xy-plane over the specified region
$f (x, y) = x y^{3} e^{x^{2} y^{2}} W h e r e R = {(x, y) | - 2 \leq x \leq - 1, - 2 \leq y \leq 0}$

Short Answer

Expert verified

The value of the volume is $- 6 - 2 e^{4} + 8 e^{16}$ cubicunits

Step by step solution

Given information

We are given a function over a region

$f (x, y) = x y^{3} e^{x^{2} y^{2}} W h e r e R = {(x, y) | - 2 \leq x \leq - 1, - 2 \leq y \leq 0}$

Find the volume

The value of the function is positive in the given region

Hence the volume becomes

$\int_{- 2}^{0} \int_{- 2}^{- 1} x y^{3} e^{x^{2} y^{2}} d x d y V = 2 \int_{- 2}^{0} {[y e^{x^{2} y^{2}}]}^{- 1}_{- 2} d y V = 2 \int_{- 2}^{0} [y e^{y^{2}} - y e^{4 y^{2}}] d y V = {[2 e^{y^{2}} - 8 e^{4 y^{2}}]}_{- 2}^{0} V = [- 6 - 2 e^{4} + 8 e^{16}]$

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Find the signed volume between the graph of the function and the xy-plane over the specified region
$f (x, y) = x y^{3} e^{x^{2} y^{2}} W h e r e R = {(x, y) | - 2 \leq x \leq - 1, - 2 \leq y \leq 0}$

Short Answer

Step by step solution

Given information

Find the volume

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Find the signed volume between the graph of the function and the xy-plane over the specified regionf(x,y)=xy3ex2y2WhereR={(x,y)|-2≤x≤-1,-2≤y≤0}

Short Answer

Step by step solution

Given information

Find the volume

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Statistics

Theoretical and Mathematical Physics

Calculus

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Find the signed volume between the graph of the function and the xy-plane over the specified region
$f (x, y) = x y^{3} e^{x^{2} y^{2}} W h e r e R = {(x, y) | - 2 \leq x \leq - 1, - 2 \leq y \leq 0}$