Chapter 13: Q. 44 (page 1079)

Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{Ω} x y^{3} d A$ , where $Ω$ is the region from Exercise 43.

Short Answer

Expert verified

$\int \int_{Ω} x y^{3} d A = 8736$

Step by step solution

Draw the region and name the vertices

The region $Ω$ is bounded by,

$y = 3 x, x y = 27, y = \frac{1}{3} x, x y = 3$

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

Consider the new set of variables defined as

$u = \frac{x}{y} v = x y$

After solving, We get that

$\sqrt{u v} = x \sqrt{\frac{v}{u}} = y$

Determine the equation of each boundary in terms of u and v.

We have,

$\sqrt{u v} = x \sqrt{\frac{v}{u}} = y$

Use these equations to determine the equation of each boundary of the region.

$A B : y = 3 x \Rightarrow u = \frac{1}{3} B C : x y = 27 \Rightarrow v = 27 C D : y = \frac{1}{3} x \Rightarrow u = 3 D A : x y = 3 \Rightarrow v = 3$

Plot these limits on a u v- plane.

Evaluate the double integral.

$\int \int_{Ω} (x y^{3}) d A = \frac{1}{2} \int_{u = \frac{1}{3}}^{u = 3} \int_{v = 3}^{v = 27} (\frac{v^{2}}{u^{2}}) d v d u \int \int_{Ω} (\frac{x^{2}}{y^{2}} + x^{2} y^{2}) d A = \frac{1}{2} \int_{u = \frac{1}{3}}^{u = 3} \frac{1}{u} (\int_{v = 3}^{v = 27} (v^{2}) d v) d u \int \int_{Ω} (\frac{x^{2}}{y^{2}} + x^{2} y^{2}) d A = \frac{1}{2} \int_{u = \frac{1}{3}}^{u = 3} \frac{1}{u} ({[\frac{v^{3}}{3}]}_{3}^{27}) d u \int \int_{Ω} (\frac{x^{2}}{y^{2}} + x^{2} y^{2}) d A = 3276 \int_{u = 1 / 3}^{u = 3} (\frac{1}{u^{2}}) d u \int \int_{Ω} (\frac{x^{2}}{y^{2}} + x^{2} y^{2}) d A = 3276 {[- \frac{1}{u}]}_{1 / 3}^{3} \int \int_{Ω} (\frac{x^{2}}{y^{2}} + x^{2} y^{2}) d A = 8736$

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Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{Ω} x y^{3} d A$ , where $Ω$ is the region from Exercise 43.

Short Answer

Step by step solution

Draw the region and name the vertices

Determine the equation of each boundary in terms of u and v.

Evaluate the double integral.

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Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.∫∫Ωxy3dA, where Ωis the region from Exercise 43.

Short Answer

Step by step solution

Draw the region and name the vertices

Determine the equation of each boundary in terms of u and v.

Evaluate the double integral.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Calculus

Decision Maths

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Evaluate the double integrals in Exercises 39–48. Use suitable transformations as necessary.
$\int \int_{Ω} x y^{3} d A$ , where $Ω$ is the region from Exercise 43.