Chapter 14.3, Problem 45E

### 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

# 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=2sin3Î¸.

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 all three petals are similar to each other. Now, let us consider the one petal which is present in the first quadrant. The region is r-simple, 0â‰¤râ‰¤2sin3Î¸

For the variation of the angle Î¸:

For r=0,

0=2sin3Î¸sin3Î¸=03Î¸=0,Ï€Î¸=0,Ï€3

From the figure, the one petal is starts from Î¸=0 to end at Î¸=Ï€3. So, the variation of Î¸ is, 0â‰¤Î¸â‰¤Ï€3

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

The total area would be the three times of the petal present in the first quadrant

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