   Chapter 7.3, Problem 35E

Chapter
Section
Textbook Problem

# Prove the formula A = 1 2 r 2 θ for the area of a sector of a circle with radius r and central angle θ . [Hint: Assume 0 < θ < π / 2 and place the centre of the circle at the origin so it has the equation x 2 + y 2 = r 2 . Then A is the sum of the area of the triangle POQ and the area of the region PQR in the figure.] To determine

To Prove: A=12r2θ

Explanation

Calculations: The line OP has the equation y=xtanθ

The circle and the line intersect when

x2(1+tan2θ)=r2x2sec2θ=r2x2=r2cos2θx=rcosθ

y=xtanθ=rcosθ.tanθ=rsinθ

Area of the triangle =12bh

Area of the triangle = 12(rcosθ)(rsinθ)

Area of the triangle =r2cosθsinθ2

Area of the blue region =rcosθrr2x2dx

We will use integral no. 30 from the table of integrals

a2u2du=u2a2u2+a22sin1ua+C

=[x2r2x2+r22sin1xr]rcosθr

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