Chapter 13: Q. 59 (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
$f (x, y) = x y W h e r e R = {(x, y) | - 2 \leq x \leq 3, - 1 \leq y \leq 5}$

Short Answer

Expert verified

The value of integral is 24cubicunits

Step by step solution

Given information

We are given an region

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

Find the volume

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

And

$\int_{R} x y d A = \int_{R_{1}} x y d A - \int_{R_{2}} x y d A = \int_{0}^{5} \int_{0}^{2} x y d x d y - \int_{- 1}^{0} \int_{- 2}^{0} x y d x d y = [\frac{x^{2} y^{2}}{4}] - [\frac{x^{2} y^{2}}{4}] = 25 - 1 = 24 c u b i c u n i t s$

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

Short Answer

Step by step solution

Given information

Find the volume

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Find the signed volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-planef(x,y)=xyWhereR={(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

Decision Maths

Pure Maths

Probability and Statistics

Applied Mathematics

Geometry

Calculus

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