Chapter 13: Q. 11 (page 1026)

Give a geometric explanation why $n \int_{0}^{2 π / n} \int_{0}^{R} r d r d θ = π R^{2}$
for any positive real number $ℝ$ and any positive integer $n$ .
Would the equation also hold for nonintegral values of $n$ ?

Short Answer

Expert verified

Area of $n$ sectors = $n \int_{0}^{2 π / n} \int_{0}^{π} r d r d θ = π R^{2} [Area of a circle]$ is shown.

Step by step solution

Given Information

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

$n \int_{0}^{2 π / n} \int_{0}^{π} r d r d θ = π R^{2}$

Calculation

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

Area of a sector $= \int_{0}^{2 π i =} \int_{0}^{R} r d r d θ$

Integrate with respect to $r$ .

$\int_{0}^{2 π / n} \int_{0}^{π} r d r d θ = \int_{0}^{2 π / n} {[\frac{r^{2}}{2}]}_{0}^{π} d θ$

$\int_{0}^{2 π^{' / n}} \int_{0}^{π} r d r d θ = \int_{0}^{2 \sin} \frac{R^{2}}{2} d θ$

$\int_{0}^{2 π / n} \int_{0}^{R} r d r d θ = \frac{R^{2}}{2} \int_{0}^{2 π / n} d θ$

Integrate with respect to $θ$ .

$\int_{0}^{2 π / n} \int_{0}^{n} r d r d θ = \frac{R^{2}}{2} [θ]_{0}^{2 π / m}$

Substitute the limits

$\int_{0}^{2 π / n} \int_{0}^{π} r d r d θ = \frac{π R^{2}}{n}$

Therefore, area of $n$ sectors

$n \int_{0}^{2 \sin} \int_{0}^{R} r d r d θ = \frac{π n R^{2}}{n}$

$n \int_{0}^{2 π / n} \int_{0}^{π} r d r d θ = π R^{2} [Area of a circle]$

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Give a geometric explanation why $n \int_{0}^{2 π / n} \int_{0}^{R} r d r d θ = π R^{2}$
for any positive real number $ℝ$ and any positive integer $n$ .
Would the equation also hold for nonintegral values of $n$ ?

Short Answer

Step by step solution

Given Information

Calculation

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Give a geometric explanation why n∫02π/n∫0Rrdrdθ=πR2for any positive real number ℝand any positive integer n.Would the equation also hold for nonintegral values of n?

Short Answer

Step by step solution

Given Information

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

Give a geometric explanation why $n \int_{0}^{2 π / n} \int_{0}^{R} r d r d θ = π R^{2}$
for any positive real number $ℝ$ and any positive integer $n$ .
Would the equation also hold for nonintegral values of $n$ ?