   Chapter 10, Problem 111RE

Chapter
Section
Textbook Problem

# Finding the Area of a Surface of Revolution In Exercises 111 and 112, 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 = 2 sin θ 0 ≤ θ ≤ π Polar axis

To determine

To calculate: The integral that represents the area of the surface formed by revolving the curve, r=1+4cosθ on the interval 0θπ2, about the polar axis. Also use the integration capabilities of a graphing utility to approximate the integral accurate to two decimal places.

Explanation

Given:

The curve, r=1+4cosθ, the interval, 0θπ2 and the axis of revolution is, polar axis.

Formula used:

The area of the surface of a polar curve is given by,

S=2παβf(θ)sinθ(f(θ))2+(f(θ))2dθ where f(θ) is the polar curve and α,β are the interval where the curve revolves.

Calculation:

Consider the given curve, r=1+4cosθ.

Now assume, f(θ)=1+4cosθ.

So, f(θ)=4sinθ.

Now, use the formula, S=2παβf(θ)sinθ(f(θ))2+(f(θ))2dθ to form the integral that calculates the surface area when the curve revolves about polar axis

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