Chapter 10.5, Problem 10E

### 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

# Finding the Area of a Polar Region In Exercises 7-18, Find the area of the region.Two petals of   r = 4 sin 3 θ

To determine

To calculate: The value of area of two petal of polar equation r=4sin3θ.

Explanation

Given:

The polar equation is r=4sin3Î¸.

Calculation:

Shaded section of two petal circumscribed by the curve r=4sin3Î¸ is shown below:

Use the following steps in Ti-83 calculator to plot the graph:

Step1: Press MODE Key and select POL mode.

Step2: Press (Y=) Key.

Step3: Press r1=4sin3Î¸

Step4: Now press the [GRAPH] key and the graph is found:

From the figure, it can be seen that pf polar region lies among 0 to 2Ï€.

The section for polar axis is drawn out for one petal is 0â‰¤Î¸â‰¤Ï€3

Substitute 4sin3Î¸ for r, Ï€3 for Î² and 0 for Î±

The area of shaded section bounded by graph of r=f(Î¸) between the radial lines Î¸=Î± and Î¸=Î².

So,

A=12âˆ«Î±Î²[f(Î¸]2.dÎ¸

Substitute these values in above formula and get,

A=12âˆ«0Ï€3[4sin3Î¸]2

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