Chapter 10.5, Problem 68E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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+cosÎ¸) and the interval is 0â‰¤Î¸â‰¤Ï€

Where the axis of revolution is polar axis.

Formula Used:

s=2Ï€âˆ«Î±Î²f(Î¸)â‹…sinÎ¸(f(Î¸))2+(fâ€²(Î¸))2dÎ¸

Calculation:

Given polar equation is r=a(1+cosÎ¸) where interval is 0â‰¤Î¸â‰¤Ï€ and axis of revolution is polar axis.

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

s=2Ï€âˆ«Î±Î²f(Î¸)â‹…sinÎ¸(f(Î¸))2+(fâ€²(Î¸))2dÎ¸

Here, r=a(1+cosÎ¸),Î±=0,Î²=Ï€

And

drdÎ¸=fâ€²(Î¸)=âˆ’asinÎ¸

Substitute these values in the above formula;

s=2Ï€âˆ«Î±Î²f(Î¸)â‹…sinÎ¸(f(Î¸))2+(fâ€²(Î¸))2dÎ¸=2Ï€âˆ«0Ï€a(1+cosÎ¸)â‹…sinÎ¸(a(1+cosÎ¸))2+(âˆ’asinÎ¸)2dÎ¸=2Ï€âˆ«0Ï€a(1+cosÎ¸)â‹…sinÎ¸(a2(1+cos2Î¸+2cosÎ¸))+(a2sin2Î¸)dÎ¸=2Ï€âˆ«0Ï€a(1+cos

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