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Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

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BuyFindarrow_forward

Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

The figure shows a sector of a circle with central angle θ. Let A(θ) be the area of the segment between the chord PR and the arc PR. Let B(θ) be the area of the triangle PQR. Find lim θ 0 + A ( θ ) / B ( θ ) .

Chapter 4.4, Problem 84E, The figure shows a sector of a circle with central angle . Let A() be the area of the segment

To determine

To evaluate: The value of limθ0+A(θ)B(θ) where θ is the central angle.

Explanation

Given:

The given figure represents a sector with central angle θ and a chord PR. The sector PR is named as A(θ) and the area of the right triangle OPR is named as B(θ)

Calculation:

Let the radius of the circle be r.

Therefore, OP=OR=r.

First find the value of A(θ).

The value of A(θ) is the difference between the area of the sector PR and the area of triangle OPR.

Area of the sector PR is given by, A(sector PR)=(θ2ππ(r)2) where θ is the central angle.

Consider the triangle OPR.

Area of the triangle OPR is 12ORPQ.

The height of the triangle OPR is, PQ.

PQ=OPsinθ=rsinθ

Hence, area of the triangle OPR is,

A(triangle OPR)=12ORPQ=12rrsinθ

Therefore, A(θ) is,

A(θ)=A(sector PR)A(triangle OPR)=(θ2ππ(r)2)12(r)(rsinθ)=(θ2r2)r2sinθ2=r22(θsinθ)

Hence, A(θ)=r22(θsinθ) (1)

Similarly, find the value of B(θ).

The value of B(θ) is the area of right angled triangle PQR with base QR.

The base QR is given by,

QR=OROQ=rrcosθ

The value of B(θ) is,

B(θ)=12(QR)(PQ)=12(rrcosθ)(rsinθ)=r22(1cosθ)(sinθ)=r22(sinθcosθsinθ)

Therefore, B(θ)=r22(sinθsinθcosθ) (2)

Evaluate the value of limθ0+A(θ)B(θ) using the equations (1) and (2),

limθ0+A(θ)B(θ)=limθ0+r22(θsinθ)r22(sinθsinθcosθ)=limθ0+θsinθsinθsinθcosθ

Obtain the value of the function as θ approaches 0+

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