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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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BuyFindarrow_forward

Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Finding the Area of a Polar Region In Exercises 97–102, find the area of the region.

Common interior of r = 4 cos θ and r = 2

To determine

To calculate: The area between the curve r=4cosθ and r=2.

Explanation

Given:

The polar equations are r=4cosθ and r=2.

Formula used:

If the function f is continuous and non-negative in the interval [α,β],0<βα2π, then the area of the region bounded by the graph of r=f(θ) between θ=α and θ=β is:

A=12αβ[f(θ)]2dθ where 0<βα2π

The formulae for integration:

cosaxdx=sinaxa+C

Trigonometric identity:

cos2θ=2cosθ1θ

Calculation:

Consider the equation, r=4cosθ and r=2. Sketch the graph of the function r=4cosθ:

At θ=0

r=4cos(0)=41=4

At θ=π4

r=4cos(π4)=42

At θ=π2

r=4cos(π2)=40=4

At θ=7π4

r=4cos(7π4)=42

At θ=2π

r=4cos2π=41=4

So, the graph of the equation is:

Compute the intersection point:

2=4cosθcosθ=12θ=cos1(12)θ=π3

So, the radial line is θ=π3. Since, both the curves are symmetric to y-axis so calculate the of the upper half-plane. For r=2 the region is between 0 to π3 and for r=4cosθ the region is between π3 to π2

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