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Chapter 14.3, Problem 44E
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### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

#### Solutions

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Section
BuyFindarrow_forward

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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+sinÎ¸.

Formula used:

The integration of:

âˆ«xndx=xn+1n+1+Câˆ«sin(ax)dx=âˆ’cos(ax)a+C.

Calculation:

From the figure, it can be observe that the radius of the curve varies from zero to 2+sinÎ¸ that means, 0â‰¤râ‰¤2+sinÎ¸

For the variation of the angle Î¸,

For r=2,

2=2+sinÎ¸sinÎ¸=0Î¸=0,2Ï€

The curve has a period of 2Ï€. The value of Î¸ is, 0â‰¤Î¸â‰¤2Ï€

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=âˆ«02Ï€âˆ«02+sinÎ¸rdrdÎ¸=âˆ«02Ï€[r22]02+sinÎ¸dÎ¸=

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