Chapter 13: Q. 39 (page 1079)

Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{R} {(x - y)}^{3} d A$ , where R is the parallelogram with vertices (0, 0), (3, 0), (5, 2), and (2, 2).

Short Answer

Expert verified

$\int \int_{R} {(x - y)}^{3} d A = \frac{81}{2}$

Step by step solution

Draw the region

Plot the given points to form the region and name the vertices.

In the region R, the equations of boundary curves are,

$A B : y = 0 B C : x - y = 3 C D : y = 2 D A : x - y = 0$

Consider the new set of variables defined as,

$u = x - y v = y$

Add the above equations,

$u + v = x$

Use these equations to determine the equation of each boundary of the region in terms of variables u and v.

$A B : v = 0 B C : u = 3 C D : v = 2 D A : u = 0$

Plot these points on UV plane,

Find the integral

Set up the double integral for the region R' between u = 0 and u = 3.

$\int \int_{R} {(x - y)}^{3} d A = \int_{0}^{3} (u^{3} \int_{0}^{2} d v) d u = \int_{0}^{3} (u^{3} {[v]}_{0}^{2}) d u = 2 \int_{0}^{3} u^{3} d u = 2 {[\frac{u^{4}}{4}]}_{0}^{3} = 2 \times \frac{81}{4} = \frac{81}{2}$

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Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{R} {(x - y)}^{3} d A$ , where R is the parallelogram with vertices (0, 0), (3, 0), (5, 2), and (2, 2).

Short Answer

Step by step solution

Draw the region

Find the integral

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Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.∫∫R(x-y)3dA, where R is the parallelogram with vertices (0, 0), (3, 0), (5, 2), and (2, 2).

Short Answer

Step by step solution

Draw the region

Find the integral

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

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Study anywhere. Anytime. Across all devices.

Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{R} {(x - y)}^{3} d A$ , where R is the parallelogram with vertices (0, 0), (3, 0), (5, 2), and (2, 2).