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Chapter 13: Q 49. (page 1066)

The region bounded below by the $x y$ -plane, bounded above by the sphere with radius $2$ and centered at the origin, and outside the cylinder with equation $x^{2} + y^{2} = 1$

Short Answer

Expert verified

The volume is $2 \sqrt{3} 蟺$ units.

Step by step solution

01

Given Information

Radius if sphere is $2$ and is centered at origin and outside cylinder with equation $x^{2} + y^{2} = 1$

02

Simplification and evaluation of limits

Relationship between cylindrical and rectangular coordinates is given by

$r = \sqrt{x^{2} + y^{2}}, \tan 胃 = \frac{y}{x}, z = z$

and

$x = r \cos 胃, y = r \sin 胃, z = z$

Equation of sphere in terms of rectangular coordinates is $x^{2} + y^{2} + z^{2} = 4$

In terms of cylindrical coordinates, equation is $r^{2} + z^{2} = 4$

$鈬� z = \sqrt{4 - r^{2}}$

Region in $x y$ plane above which lies the surface is

$x^{2} + y^{2} = 1 鈬� x^{2} + y^{2} = 2 y$

And

$r^{2} = 2 r \sin 胃, 鈬� r = 2 \sin 胃$

Limits in cylindrical coordinates are

$2 \sin 胃鈮� z 鈮� \sqrt{4 - r^{2}}, 1 鈮� r 鈮� 2, 0 鈮� 胃鈮� 蟺$

03

Evaluating the volume

Required volume is given by

$V = {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{2} {鈭�}_{2 \sin 胃}^{\sqrt{4 - r^{2}}} r d z d r d 胃$

$鈬� V = {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{2} r (\sqrt{4 - r^{2}} - 2 \sin 胃) d r d 胃$

$鈬� V = \frac{1}{2} {鈭�}_{0}^{2 蟺} {[\frac{{(4 - r^{2})}^{\frac{3}{2}}}{- \frac{3}{2}}]}_{1}^{2} d 胃 - {鈭�}_{0}^{2 蟺} 2 \sin 胃 {(r)}_{1}^{2} d 胃$

$= \frac{1}{2} {鈭�}_{0}^{2 蟺} (- \frac{2}{3}) (0 - 3 \sqrt{3}) d 胃 + 2 {(\cos 胃)}_{0}^{2 蟺}$

Simplifying we get

$= 2 \sqrt{3} 蟺$

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Most popular questions from this chapter

Find the volume between the graph of the given function and the xy-plane over the specified rectangle in the xy-plane

$f (x, y) = \frac{y}{x} e^{y} W j e r e R = {(x, y) | 1 鈮� x 鈮� e, 0 鈮� y 鈮� 2}$

Evaluate the triple integrals over the specified rectangular solid region.

$\begin{aligned} {鈭�}_{R} 鈥� \ln 鈦� (x y z^{2}) d V, where \\ R = {(x, y, z) 鈭� 1 鈮� x 鈮� 3, 1 鈮� y 鈮� e, and 1 鈮� z 鈮� 2} \end{aligned}$

Describe the three-dimensional region expressed in each iterated integral:

${鈭�}_{鈭� 2}^{4} 鈥� {鈭�}_{2}^{6} 鈥� {鈭�}_{0}^{5} 鈥� f (x, y, z) d y d x d z$

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

${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - x^{2}}}^{\sqrt{9 - x^{2}}} {鈭�}_{- \sqrt{9 - x^{2} - y^{2}}}^{\sqrt{9 - x^{2} - y^{2}}} f (x, y, z) d z d y d x$

Explain why ${鈭�}_{i = 1}^{l} {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} i j^{2} k^{3} = {鈭�}_{j = 1}^{m} {鈭�}_{k = 1}^{n} {鈭�}_{i = 1}^{l} i j^{2} k^{3}$ .

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