Q. 44 Evaluate the double integrals in... [FREE SOLUTION]

91影视

Chapter 13: Q. 44 (page 1079)

Evaluate the double integrals in Exercises 39鈥�48. Use suitable transformations as necessary.
$鈭� {鈭�}_{惟} x y^{3} d A$ , where $惟$ is the region from Exercise 43.

Short Answer

Expert verified

$鈭� {鈭�}_{惟} x y^{3} d A = 8736$

Step by step solution

Draw the region and name the vertices

The region $惟$ 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.

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

Plot these limits on a u v- plane.

Evaluate the double integral.

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

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Evaluate the double integrals in Exercises 39鈥�48. Use suitable transformations as necessary.
$鈭� {鈭�}_{惟} x y^{3} d A$ , where $惟$ 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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Discrete Mathematics

Probability and Statistics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

91影视

Evaluate the double integrals in Exercises 39鈥�48. Use suitable transformations as necessary.鈭�鈭�惟xy3dA, where 惟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.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Discrete Mathematics

Probability and Statistics

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

Evaluate the double integrals in Exercises 39鈥�48. Use suitable transformations as necessary.
$鈭� {鈭�}_{惟} x y^{3} d A$ , where $惟$ is the region from Exercise 43.