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Chapter 13: Q. 29 (page 1083)

Evaluating triple integrals: Each of the triple integrals that follows represents the volume of a solid. Sketch the solid and evaluate the integral.
${鈭�}_{- 2}^{2} {鈭�}_{0}^{\sqrt{4 - x^{2}}} {鈭�}_{0}^{y} d z d y d x$

Short Answer

Expert verified

${鈭�}_{- 2}^{2} {鈭�}_{0}^{\sqrt{4 - x^{2}}} {鈭�}_{0}^{y} d z d y d x = \frac{16}{3}$

Step by step solution

01

Given

We have to find the volume by using the triple integral.

{鈭�}_{- 2}^{2} {鈭�}_{0}^{\sqrt{4 - x^{2}}} {鈭�}_{0}^{y} d z d y d x

02

Evaluate the integral

$V = {鈭�}_{- 2}^{2} {鈭�}_{0}^{\sqrt{4 - x^{2}}} [{鈭�}_{0}^{y} d z] d y d x V = {鈭�}_{- 2}^{2} [{鈭�}_{0}^{\sqrt{4 - x^{2}}} y d y] d x V = {鈭�}_{- 2}^{2} {[\frac{y^{2}}{2}]}_{0}^{\sqrt{4 - x^{2}}} d x V = \frac{1}{2} {鈭�}_{- 2}^{2} (4 - x^{2}) d x V = 2 脳 \frac{1}{2} {鈭�}_{0}^{2} (4 - x^{2}) d x V = {[4 x - \frac{x^{3}}{3}]}_{0}^{2} V = [8 - \frac{8}{3}] V = \frac{16}{3} cubic units$

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Most popular questions from this chapter

Evaluate the sums in Exercises 23鈥�28.

${鈭�}_{j = 1}^{3} {鈭�}_{k = 1}^{2} k^{j}$

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

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

Evaluate each of the double integrals in Exercises 37鈥�54 as iterated integrals.

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

Let $惟$ be a lamina in the xy-plane. Suppose $惟$ is composed of two non-overlapping lamin $惟_{1}$ and $惟_{2}$ , as follows:

Show that if the masses and centers of masses of $惟_{1}$ and $惟_{2}$ are $m_{1}$ and $m_{2},$ and $({\overset{炉}{x}}_{1}, {\overset{炉}{y}}_{1}) and ({\overset{炉}{x}}_{2}, {\overset{炉}{y}}_{2})$ respectively, then the center of mass of $惟$ is $(\overset{炉}{x}, \overset{炉}{y}),$ where

$\overset{炉}{x} = \frac{m_{1} {\overset{炉}{x}}_{1} + m_{2} {\overset{炉}{x}}_{2}}{m_{1} + m_{2}} and \overset{炉}{y} = \frac{m_{1} {\overset{炉}{y}}_{1} + m_{2} {\overset{炉}{y}}_{2}}{m_{1} + m_{2}}$

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.

${鈭�}_{惟} 鈥� x 蚁 (x, y, z) d V$

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