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Chapter 13: Q. 43 (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.
${鈭�}_{0}^{2 蟺} {鈭�}_{0}^{3} (6 r - 2 r^{2}) d r d 胃$

Short Answer

Expert verified

The value of integral is

${鈭�}_{0}^{2 蟺} {鈭�}_{0}^{3} (6 r - 2 r^{2}) d r d 胃 = 18 蟺$

Step by step solution

01

Given information

The expression is

I = {鈭�}_{0}^{2 蟺} {鈭�}_{0}^{3} (6 r - 2 r^{2}) d r d 胃

02

Calculation

Here, $r = 0, r = 3$ and $胃 = 0, 胃 = 2 蟺$

Integrate with respect to $r$ first,

$I = {鈭�}_{0}^{2 蟺} {[\frac{6 r^{2}}{2} - \frac{2 r^{3}}{3}]}_{0}^{3} d 胃 [鈭� x^{n} d x = \frac{x^{n + 1}}{n + 1} + C]$

Put the limits

localid="1651390685586" $I = {鈭�}_{0}^{2 蟺} [\frac{6 (3)^{2}}{2} - \frac{2 (3)^{3}}{3}] d 胃 I = {鈭�}_{0}^{2 蟺} 9 d 胃$

Now integrate with respect to $胃$

$I = 9 [胃]_{0}^{2 蟺} I = 18 蟺$

Thus, the value of integral is

${鈭�}_{0}^{2 蟺} {鈭�}_{0}^{3} (6 r - 2 r^{2}) d r d 胃 = 18 蟺$

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Most popular questions from this chapter

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}.

Assume that the density ofR is uniform throughout.

(a) Without using calculus, explain why the center of mass is (2, 3/2, 1).

(b) Verify that the center of mass is (2, 3/2, 1), using the appropriate integral expressions.

Evaluate the triple integrals over the specified rectangular solid region.

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

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

$f (x, y) = x y W h e r e R = {(x, y) | - 2 鈮� x 鈮� 3, - 1 鈮� y 鈮� 5}$

$Express the sum 3 e^{4} + 3 e^{9} + 3 e^{16} + 4 e^{4} + 4 e^{9} + 4 e^{16} using double - summation notation .$

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.

${鈭�}_{惟} 鈥� (x^{2} + z^{2}) 蚁 (x, y, z) d V$

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