   Chapter 14.3, Problem 44E

Chapter
Section
Textbook Problem

AreaIn Exercises 41–46, use a double integral to find the area of the shaded region. To determine

To calculate: The area of the shaded region of the figure using the double integration. Explanation

Given:

The figure is

The expression of r is r=2+sinθ.

Formula used:

The integration of:

xndx=xn+1n+1+Csin(ax)dx=cos(ax)a+C.

Calculation:

From the figure, it can be observe that the radius of the curve varies from zero to 2+sinθ that means, 0r2+sinθ

For the variation of the angle θ,

For r=2,

2=2+sinθsinθ=0θ=0,2π

The curve has a period of 2π. The value of θ is, 0θ2π

From obtained data, setup the double integral in polar coordinates for the area of the shaded region:

The expression for the double integral is,

A=02π02+sinθrdrdθ=02π[r22]02+sinθdθ=

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