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Chapter 13: Q. 48 (page 1079)

Evaluate the double integrals in Exercises 39鈥�48. Use suitable transformations as necessary.
$鈭� {鈭�}_{惟} x^{2} (1 - \frac{1}{y^{2}}) d A$ , where $惟$ is the region from Exercise 47.

Short Answer

Expert verified

$鈭� {鈭�}_{惟} x^{2} (1 - \frac{1}{y^{2}}) d A = (\frac{3}{4} \ln (\frac{4}{3}) - \frac{3}{8})$

Step by step solution

01

Draw the region and name the vertices

The region $惟$ 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,

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

02

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.

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

Plot these limits on u v plane.

03

Evaluate the double integral.

Set up the double integral,

鈭� {鈭�}_{惟} x^{2} (1 - \frac{1}{y^{2}}) d A = \frac{1}{4} {鈭�}_{u = 1}^{u = 4} {鈭�}_{v = 0}^{v = \frac{1}{2}} (\frac{1}{1 - v^{2}} - 1) (2 v) d v d u 鈭� {鈭�}_{惟} x^{2} (1 - \frac{1}{y^{2}}) d A = \frac{1}{4} {鈭�}_{u = 1}^{u = 4} {鈭�}_{v = 0}^{v = \frac{1}{2}} (\frac{2 v}{1 - v^{2}} - 2 v) d v d u 鈭� {鈭�}_{惟} x^{2} (1 - \frac{1}{y^{2}}) d A = \frac{1}{4} {鈭�}_{u = 1}^{u = 4} ({[- \ln (1 - v^{2})]}_{0}^{1 / 2} - {[v^{2}]}_{0}^{1 / 2}) 鈭� {鈭�}_{惟} x^{2} (1 - \frac{1}{y^{2}}) d A = (\frac{1}{4} \ln (\frac{4}{3}) - \frac{1}{8}) {鈭�}_{u = 1}^{u = 4} d u 鈭� {鈭�}_{惟} x^{2} (1 - \frac{1}{y^{2}}) d A = 3 脳 (\frac{1}{4} \ln (\frac{4}{3}) - \frac{1}{8}) 鈭� {鈭�}_{惟} x^{2} (1 - \frac{1}{y^{2}}) d A = (\frac{3}{4} \ln (\frac{4}{3}) - \frac{3}{8})

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Most popular questions from this chapter

Describe the three-dimensional region expressed in each iterated integral in Exercises 35鈥�44.

${鈭�}_{0}^{3} {鈭�}_{0}^{3} {鈭�}_{0}^{3 - y} f (x, y, z) d z d y d x$

Use Definition $13.4$ to evaluate the double integrals in Exercises $29 鈥� 32$ .

${鈭�}_{R} x y d A$

where

R = {(x, y) 鈭� 0 鈮� x 鈮� 2 & 1 鈮� y 鈮� 4}

In Exercises 61鈥�64, let R be the rectangular solid defined by

$R = \{(x, y, z) | 0 鈮� a_{1} 鈮� x 鈮� a_{2}, 0 鈮� b_{1} 鈮� y 鈮� b_{2}, 0 鈮� c_{2} 鈮� z 鈮� c_{2}\} .$

Assume that the density of R is uniform throughout.

(a) Without using calculus, explain why the center of

mass is $(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}, \frac{c_{1} + c_{2}}{2}) .$

(b) Verify that $(\frac{a_{1} + a_{2}}{2}, \frac{b_{1} + b_{2}}{2}, \frac{c_{1} + c_{2}}{2})$ is the center of mass by using the appropriate integral expressions.

$L e t R = \{(x, y, z) | a_{1} 猢� x 猢� a_{2}, b_{1} 猢� y 猢� b_{2}, c_{1} 猢� z 猢� c_{2}\} . P r o v e t h a t : {鈭�}_{R} d V = (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) . W h a t i s t h e r e l a t i o n s h i p b e t w e e n R a n d t h e p r o d u c t (a_{2} - a_{1}) (b_{2} - b_{1}) (c_{2} - c_{1}) .$

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$鈭� {鈭�}_{R} e^{x y} d A w h e r e R = \{(x, y) I 1 鈮� x 鈮� 2 a n d 1 鈮� y 鈮� 3\}$

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