Chapter 13: Q. 60 (page 1005)

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = - 2 x^{2} y^{3} W h e r e R = {(x, y) | - 2 \leq x \leq 3, - 1 \leq y \leq 5}$

Short Answer

Expert verified

The value of the integral is $1686.66 c u b i c u n i t s$

Step by step solution

Given information

We are given a rectangular region

$f (x, y) = - 2 x^{2} y^{3} W h e r e R = {(x, y) | - 2 \leq x \leq 3, - 1 \leq y \leq 5}$

Find the volume

We have,

$f (x, y) > 0 w h e n R_{1} = {(x, y) | - 2 \leq x \leq 0, - 1 \leq y \leq 0} A n d f (x, y) < 0 w h e n R_{2} = {(x, y) | 0 \leq x \leq 3, 0 \leq y \leq 5}$

Also we have,

$\int_{R} - 2 x^{2} y^{3} d A = \int_{R_{1}} - 2 x^{2} y^{3} d A - \int_{R_{2}} - 2 x^{2} y^{3} d A = \int_{- 1}^{0} \int_{- 2}^{0} - 2 x^{2} y^{3} d x d y - \int_{0}^{3} \int_{0}^{5} - 2 x^{2} y^{3} d x d y = [\frac{- 2 x^{3} y^{4}}{12}] - [\frac{- 2 x^{3} y^{4}}{12}] = - \frac{16}{12} + [\frac{20250}{12}] = 1686.66 c u b i c u n i t s$

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Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = - 2 x^{2} y^{3} W h e r e R = {(x, y) | - 2 \leq x \leq 3, - 1 \leq y \leq 5}$

Short Answer

Step by step solution

Given information

Find the volume

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Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-planef(x,y)=-2x2y3WhereR={(x,y)|-2≤x≤3,-1≤y≤5}

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

Calculus

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Study anywhere. Anytime. Across all devices.

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane
$f (x, y) = - 2 x^{2} y^{3} W h e r e R = {(x, y) | - 2 \leq x \leq 3, - 1 \leq y \leq 5}$