   Chapter 10.5, Problem 75E

Chapter
Section
Textbook Problem

# Approximating Area Consider the circle r = 8 cos θ .(a) Find the area of the circle.(b) Complete the table for the areas A of the sectors of the circle between θ = 0 and the values of θ in the table. θ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 A (c) Use the table in part (b) to approximate the values of θ for which the sector of the circle composes 1 4 , 1 2 , and 3 4 of the total area of the circle.(d) Use a graphing utility to approximate, to two decimal places, the angles θ for which the sector of the circle composes 1 4 , 1 2 , and 3 4 of the total area of the circle.(e) Do the results of part (d) depend on the radius of the circle? Explain.

(a)

To determine

To Calculate: The value of the area of circle of equation r=8cosθ.

Explanation

Given:

Equation of circle r=8cosθ.

Formula Used:

A=12αβ[f(θ)]2dθ

Calculation:

To find the limits for θ put r=0.

Then 0=8cosθ;

Therefore,

cosθ=cosπ2θ=π2

So, the area of region lies between the lines θ=π2 to θ=π2

Therefore, area of region is given as:

A=12π2π2[8cosθ]2dθ=12

(b)

To determine

To Calculate: To Complete the tables for the areas A of sectors of circle between θ=0 and

values of θ in the table.

(c)

To determine
Which of the sector of circle composes 14,12 and 34 of total area of circle of equation r=8cosθ by using the table in part (b).

(d)

To determine

To Calculate: Which of the sector of circle composes 14,12 and 34 of total area of circle of equation r=8cosθ by using the graphing utility.

(e)

To determine
If the results of part d depend upon the radius of the circle

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