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Give a geometric explanation why n02π/n0Rrdrdθ=πR2

for any positive real number and any positive integer n.

Would the equation also hold for nonintegral values of n?

Short Answer

Expert verified

Area of nsectors = n02π/n0πrdrdθ=πR2[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

n02π/n0πrdrdθ=πR2

02

Calculation

Suppose a circle of radius Ris divided into nsectors of equal areas. Each sector will subtend an angle of 2πn at the center of circle. Area of a sector can be calculated in polar form as an integral.

Area of a sector =02πi=0Rrdrdθ

Integrate with respect to r.

02π/n0πrdrdθ=02π/nr220πdθ

02π'/n0πrdrdθ=02sinR22dθ

02π/n0Rrdrdθ=R2202π/ndθ

Integrate with respect to θ.

02π/n0nrdrdθ=R22[θ]02π/m

Substitute the limits

02π/n0πrdrdθ=πR2n

Therefore, area ofnsectors

n02sin0Rrdrdθ=πnR2n

n02π/n0πrdrdθ=πR2[Area of a circle]

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