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Chapter 13: Q. 44 (page 1027)

Each of the integrals or integral expressions in Exercises 39-46 represents the volume of a solid in ${鈩�}^{3}$ . Use polar coordinates to describe the solid, and evaluate the expressions.
$h {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{R} (r - \frac{r^{2}}{R}) d r d 胃$

Short Answer

Expert verified

The value of integral is

$h {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{R} (r - \frac{r^{2}}{R}) d r d 胃 = \frac{蟺 h R^{2}}{3}$

Step by step solution

01

Given information

The expression is $I = h {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{R} (r - \frac{r^{2}}{R}) d r d 胃$

02

Calculation

Here, $r = 0, r = R$ and $胃 = 0, 胃 = 2 蟺$

Integrate with respect to r first,

$I = h {鈭�}_{0}^{2 蟺} {[\frac{r^{2}}{2} - \frac{r^{3}}{3 R}]}_{0}^{R} d 胃 [鈭� x^{n} d x = \frac{x^{n + 1}}{n + 1} + C]$

Put the limits

$I = h {鈭�}_{0}^{2 蟺} [\frac{R^{2}}{2} - \frac{R^{3}}{3 R}] d 胃$

localid="1651390798747" $I = h {鈭�}_{0}^{2 蟺} \frac{R^{2}}{6} d 胃$

Now integrate with respect to $胃$

$I = \frac{h R^{2}}{6} [胃]_{0}^{2 蟺} I = \frac{h R^{2}}{6} [2 蟺]$

$I = \frac{蟺 h R^{2}}{3}$

Thus, the value of integral is

localid="1651390833501" $h {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{R} (r - \frac{r^{2}}{R}) d r d 胃 = \frac{蟺 h R^{2}}{3}$

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Most popular questions from this chapter

What is the difference between a double integral and an iterated integral?

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}

Evaluate Each of the integrals in exercises 33-36 as an iterated integral and then compare your answer with thoise you found in exercise 29-32

$鈭� {鈭�}_{R} x y d A W h e r e R = {(x, y) | 0 鈮� x 鈮� 2, 1 鈮� y 鈮� 4}$

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

localid="1649936867482" ${鈭�}_{x} x^{2} y d A$

where $R = {(x, y) 鈭� 1 鈮� x 鈮� 3 and 0 鈮� y 鈮� 2}$

In Exercises 57鈥�60, let R be the rectangular solid defined by

R = {(x, y, z) | 0 鈮� x 鈮� 4, 0 鈮� y 鈮� 3, 0 鈮� z 鈮� 2}.

Assuming that the density at each point in R is proportional to the distance of the point from the xy-plane, find the moment of inertia about the x-axis and the radius of gyration about the x-axis.

See all solutions

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