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Chapter 13: Q 52. (page 1067)

Find the volume of solid of region bounded above by the sphere with equation $蚁 = 2$ and bounded below by the cone with equation $蠒 = \frac{蟺}{3}$ .

Short Answer

Expert verified

The required volume is $V = \frac{8}{3} 蟺$ units.

Step by step solution

01

Given Information

The given equations are $蚁 = 2$ and $蠒 = \frac{蟺}{3}$ .

02

Evaluation of limits

We know that

$x = 蚁 \sin 蠒 \cos 胃, y = 蚁 \sin 蠒 \sin 胃, z = 蚁 \cos 蠒$

and

$蚁 = \sqrt{x^{2} + y^{2} + z^{2}}, \tan 胃 = \frac{y}{x}, \cos 蠒 = \frac{z}{蚁}, d x d y d z = 蚁^{2} \sin 蠒 d 蚁 d 蠒 d 胃$

Limits of spherical coordinates are

$0 < 胃 < 2 蟺, 0 < 蠒 < \frac{蟺}{3}, 0 < 蚁 < 2$

To find the volume as per given conditions, we will use spherical coordinates.

03

Calculation of Volume

Required Volume is $V = {鈭�}_{V} d x d y d z$

$V = {鈭�}_{蠒 = 0}^{蟺 / 3} {鈭�}_{蚁 = 0}^{2} {鈭�}_{胃 = 0}^{2 蟺} 蚁^{2} \sin 蠒 d 蚁 d 蠒 d 胃$

$V = {鈭�}_{蠒 = 0}^{蟺 / 3} \sin 蠒 d 蠒 {鈭�}_{蚁 = 0}^{2} 蚁^{2} d 蚁 {鈭�}_{胃 = 0}^{2 蟺} d 胃$

$V = \{{(- \cos 蠒)|}_{0}^{蟺 / 3}\} \{{\frac{蚁^{3}}{3}|}_{蚁 = 0}^{蚁 = 2}\} \{{胃|}_{胃 = 0}^{2 蟺}\}$

Application of limits yields

$V = \{(1 - \frac{1}{2})\} \{\frac{2^{3}}{3}\} {2 蟺}$

Hence, $V = \frac{8}{3} 蟺$ units

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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.

${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - y^{2}}}^{\sqrt{9 - y^{2}}} {鈭�}_{0}^{9 - y^{2} - z^{2}} f (x, y, z) d x d z d y$

Identify the quantities determined by the integral expressions in Exercises 19鈥�24. If x, y, and z are all measured in centimeters and 蚁(x, y,z) is a density function in grams per cubic centimeter on the three-dimensional region , give the units of the expression.

${鈭�}_{惟} 鈥� z 蚁 (x, y, z) d V$

Evaluate each of the integrals in exercise 33-36 as iterated integrals and then compare your answers with those you found in exercise 29-32

$鈭� {鈭�}_{R} x^{2} y^{3} d A W h e r e R = {(x, y) | 1 鈮� x 鈮� 3, 0 鈮� y 鈮� 2}$

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

${鈭�}_{- 1}^{5} {鈭�}_{- 3}^{2} {鈭�}_{4}^{8} f (x, y, z) d z d x d y$

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

${鈭�}_{- 3}^{3} {鈭�}_{- \sqrt{9 - z^{2}}}^{\sqrt{9 - z^{2}}} {鈭�}_{0}^{3 - \sqrt{x^{2} + z^{2}}} f (x, y, z) d y d x d z$

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