   Chapter 10, Problem 112RE

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 2 θ = π 2

To determine

To calculate: The area of the surface formed by revolving the polar equation of a curve given as, r=2sinθ over the interval 0θπ2 about the angle given as, θ=π2.

Explanation

Given:

The polar equation is r=2sinθ and interval is 0θπ2. The axis of revolution is θ=π2.

Formula used:

The area of the surface formed by revolving the curve given as, r=f(θ) in the interval αθβ, about the polar axis is given by;

S=2παβf(θ)sinθ[f(θ)]2+[f(θ)]2dθ

The derivative of ddθ(sinθ)=cosθ.

The trigonometric identity sin2x+cos2x=1, sin2θ=1cos2θ2.

Calculation:

The polar equation is;

r=2sinθ

The derivative of the polar equation with respect to θ is;

drdθ=ddθ(2sinθ)=2cosθ

Then, the surface area formed is;

S=2π0π2(2sinθ)cosθ[2sinθ]2+[2cosθ]2dθ=2π0π2(2cosθsinθ)4sin2θ+4cos2θdθ=4π0π2(cos<

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