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Chapter 13: Q. 42 (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.
${鈭�}_{0}^{蟺} {鈭�}_{0}^{R} (R^{2} r - r^{3}) d r d 胃$

Short Answer

Expert verified

The value of integral is ${鈭�}_{0}^{z} {鈭�}_{0}^{蟺} (R^{2} r - r^{3}) d r d 胃 = \frac{蟺 R^{4}}{4}$

Step by step solution

01

Given information

The expression is $I = {鈭�}_{0}^{蟺} {鈭�}_{0}^{R} (R^{2} r - r^{3}) d r d 胃$

02

Calculation

The objective of this problem is to use polar coordinates to describe the solid and evaluate the integral expression.

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

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

Integrate with respect to r first,

$I = {鈭�}_{0}^{蟺} {[\frac{R^{2} r^{2}}{2} - \frac{r^{4}}{4}]}_{0}^{R} d 胃 [鈭� x^{n} d x = \frac{x^{n + 1}}{n + 1} + C]$

Substitute the limits

$I = {鈭�}_{0}^{蟺} [\frac{R^{4}}{2} - \frac{R^{4}}{4}] d 胃$

I = i n t_{0}^{蟺} [\frac{R^{4}}{4}] d 胃

Now integrate with respect to $胃$

I = \frac{R^{4}}{4} [胃]_{0}^{蟺} I = \frac{蟺 R^{4}}{4}

Thus, the value of integral is

${鈭�}_{0}^{z} {鈭�}_{0}^{蟺} (R^{2} r - r^{3}) d r d 胃 = \frac{蟺 R^{4}}{4}$

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Most popular questions from this chapter

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

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

Use the lamina from Exercise 64, but assume that the density is proportional to the distance from the x-axis.

Evaluate the triple integrals over the specified rectangular solid region.

$\begin{aligned} {鈭�}_{R} 鈥� (x + 2 y + 3 z) d V, where \\ R = {(x, y, z) 鈭� 0 鈮� x 鈮� 4, 1 鈮� y 鈮� 5, and 2 鈮� z 鈮� 7} \end{aligned}$

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

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

Discuss the similarities and differences between the definition of the double integral found in Section $13.1$ and the definition of the triple integral found in this section.

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