Chapter 13: Q. 42 (page 1079)

Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{R} (3 x^{2} - 5 x y - 2 y^{2}) d A$ , where R is the parallelogram from Exercise 41

Short Answer

Expert verified

$\int \int_{R} (3 x^{2} - 5 x y - 2 y^{2}) d A = - \frac{343}{4}$

Step by step solution

Draw the region

Step 1: Draw the region

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

Consider the new set of variables defined as,

$u = x - 2 y v = 3 x + y$

From the above equations,

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Plot these limits on u v plane,

Find the integral

Use the antiderivative to evaluate all integrals,

$\int \int_{R} (3 x^{2} - 5 x y - 2 y^{2}) d A = \frac{7}{2} \int_{- 7}^{0} u d u = \frac{7}{2} {[\frac{u^{2}}{2}]}_{- 7}^{0} = \frac{7}{2} (- \frac{7^{2}}{2}) = - \frac{343}{4}$

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Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{R} (3 x^{2} - 5 x y - 2 y^{2}) d A$ , where R is the parallelogram from Exercise 41

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(3x2-5xy-2y2)dA, where R is the parallelogram from Exercise 41

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

Statistics

Pure Maths

Geometry

Probability and Statistics

Calculus

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{R} (3 x^{2} - 5 x y - 2 y^{2}) d A$ , where R is the parallelogram from Exercise 41