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Chapter 13: Q. 11 (page 1026)

Give a geometric explanation why $n {鈭�}_{0}^{2 蟺 / n} {鈭�}_{0}^{R} r d r d 胃 = 蟺 R^{2}$
for any positive real number $鈩�$ and any positive integer $n$ .
Would the equation also hold for nonintegral values of $n$ ?

Short Answer

Expert verified

Area of $n$ sectors = $n {鈭�}_{0}^{2 蟺 / n} {鈭�}_{0}^{蟺} r d r d 胃 = 蟺 R^{2} [Area of a circle]$ is shown.

Step by step solution

01

Given Information

The objective of this problem is to give the geometric explanation why

$n {鈭�}_{0}^{2 蟺 / n} {鈭�}_{0}^{蟺} r d r d 胃 = 蟺 R^{2}$

02

Calculation

Suppose a circle of radius $R$ is divided into $n$ sectors of equal areas. Each sector will subtend an angle of $\frac{2 蟺}{n}$ at the center of circle. Area of a sector can be calculated in polar form as an integral.

Area of a sector $= {鈭�}_{0}^{2 蟺 i =} {鈭�}_{0}^{R} r d r d 胃$

Integrate with respect to $r$ .

${鈭�}_{0}^{2 蟺 / n} {鈭�}_{0}^{蟺} r d r d 胃 = {鈭�}_{0}^{2 蟺 / n} {[\frac{r^{2}}{2}]}_{0}^{蟺} d 胃$

${鈭�}_{0}^{2 蟺^{' / n}} {鈭�}_{0}^{蟺} r d r d 胃 = {鈭�}_{0}^{2 \sin} \frac{R^{2}}{2} d 胃$

${鈭�}_{0}^{2 蟺 / n} {鈭�}_{0}^{R} r d r d 胃 = \frac{R^{2}}{2} {鈭�}_{0}^{2 蟺 / n} d 胃$

Integrate with respect to $胃$ .

${鈭�}_{0}^{2 蟺 / n} {鈭�}_{0}^{n} r d r d 胃 = \frac{R^{2}}{2} [胃]_{0}^{2 蟺 / m}$

Substitute the limits

${鈭�}_{0}^{2 蟺 / n} {鈭�}_{0}^{蟺} r d r d 胃 = \frac{蟺 R^{2}}{n}$

Therefore, area of $n$ sectors

$n {鈭�}_{0}^{2 \sin} {鈭�}_{0}^{R} r d r d 胃 = \frac{蟺 n R^{2}}{n}$

$n {鈭�}_{0}^{2 蟺 / n} {鈭�}_{0}^{蟺} r d r d 胃 = 蟺 R^{2} [Area of a circle]$

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Most popular questions from this chapter

In Exercises 45鈥�52, rewrite the indicated integral with the specified order of integration.

Exercise 42 with the order dy dx dz.

$How many summands are in {鈭�}_{j = j_{0}}^{m} {鈭�}_{k = k_{0}}^{n} j^{k} ?$

Evaluate the sums in Exercises $23 鈥� 28$ .

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

Explain why it would be difficult to evaluate the double integrals in Exercises 18 and 19 as iterated integrals.

$鈭� {鈭�}_{R} \cos (x y) d A w h e r e R = \{(x, y) I \frac{蟺}{4} 鈮� x 鈮� \frac{蟺}{2} a n d \frac{蟺}{2} 鈮� y 鈮� 蟺\}$

Evaluate the triple integrals over the specified rectangular solid region.

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

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