Chapter 13: Q. 47 (page 1079)

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

Short Answer

Expert verified

$\int \int_{Ω} \frac{x^{3}}{y^{3}} d A = \frac{3}{4} \ln (\frac{4}{3}) - \frac{3}{8}$

Step by step solution

Draw the region and name the vertices

The region $Ω$ is bounded by,

$x = 0, y^{2} - x^{2} = 4, y = 2 x, y^{2} - x^{2} = 1$

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

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

$A B : y^{2} - x^{2} = 1 B C : y = 2 x C D : y^{2} - x^{2} = 4 D A : x = 0$

Consider the new set of variables defined as,

$u = y^{2} - x^{2} v = \frac{x}{y}$

After solving we get that,

$\sqrt{\frac{u}{1 - v^{2}}} = y \sqrt{\frac{u v^{2}}{1 - v^{2}}} = x$

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

We have,

$\sqrt{\frac{u}{1 - v^{2}}} = y \sqrt{\frac{u v^{2}}{1 - v^{2}}} = x$

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

$A B : y^{2} - x^{2} = 1 \Rightarrow u = 1 B C : y = 2 x \Rightarrow v = \frac{1}{2} C D : y^{2} - x^{2} = 4 \Rightarrow u = 4 D A : x = 0 \Rightarrow v = 0$

Plot these limits on u v plane.

Evaluate the double integral.

Set up the double integral,

$\int \int_{Ω} (\frac{x^{3}}{y^{3}}) d A = \frac{1}{4} \int_{u = 1}^{u = 4} \int_{v = 0}^{v = \frac{1}{2}} (\frac{1}{1 - v^{2}} - 1) (2 v) d v d u \int \int_{Ω} (\frac{x^{3}}{y^{3}}) d A = \frac{1}{4} \int_{u = 1}^{u = 4} \int_{v = 0}^{v = \frac{1}{2}} (\frac{2 v}{1 - v^{2}} - 2 v) d v d u \int \int_{Ω} (\frac{x^{3}}{y^{3}}) d A = \frac{1}{4} \int_{u = 1}^{u = 4} ({[- \ln (1 - v^{2})]}_{0}^{1 / 2} - {[v^{2}]}_{0}^{1 / 2}) \int \int_{Ω} (\frac{x^{3}}{y^{3}}) d A = (\frac{1}{4} \ln (\frac{4}{3}) - \frac{1}{8}) \int_{u = 1}^{u = 4} d u \int \int_{Ω} (\frac{x^{3}}{y^{3}}) d A = 3 \times (\frac{1}{4} \ln (\frac{4}{3}) - \frac{1}{8}) \int \int_{Ω} (\frac{x^{3}}{y^{3}}) d A = (\frac{3}{4} \ln (\frac{4}{3}) - \frac{3}{8})$

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

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.∫∫Ωx3y3dA, where Ωis the following region:

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

Applied Mathematics

Statistics

Decision Maths

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

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