   Chapter 14.3, Problem 45E

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=2sin3θ.

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 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, 0r2sin3θ

For the variation of the angle θ:

For r=0,

0=2sin3θsin3θ=03θ=0,πθ=0,π3

From the figure, the one petal is starts from θ=0 to end at θ=π3. So, the variation of θ is, 0θπ3

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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