   Chapter 10.5, Problem 68E

Chapter
Section
Textbook Problem

Finding the Area of a Surface of Revolution In Exercises 65-68, find the area of the surface formed by revolving the polar equation over the given interval about the given line.Polar Equation Interval Axis of Revolution r = a ( 1 + cos θ ) 0 ≤ θ ≤ π Polar axis

To determine

To calculate: The surface area formed by revolving the polar equation r=a(1+cosθ) over the given interval 0θπ about the line.

Explanation

Given:

The polar equation is r=a(1+cosθ) and the interval is 0θπ

Where the axis of revolution is polar axis.

Formula Used:

s=2παβf(θ)sinθ(f(θ))2+(f(θ))2dθ

Calculation:

Given polar equation is r=a(1+cosθ) where interval is 0θπ and axis of revolution is polar axis.

Now, area of surface will be given by below formula;

s=2παβf(θ)sinθ(f(θ))2+(f(θ))2dθ

Here, r=a(1+cosθ),α=0,β=π

And

drdθ=f(θ)=asinθ

Substitute these values in the above formula;

s=2παβf(θ)sinθ(f(θ))2+(f(θ))2dθ=2π0πa(1+cosθ)sinθ(a(1+cosθ))2+(asinθ)2dθ=2π0πa(1+cosθ)sinθ(a2(1+cos2θ+2cosθ))+(a2sin2θ)dθ=2π0πa(1+cos

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