   Chapter 10.5, Problem 14E

Chapter
Section
Textbook Problem

# Finding the Area of a Polar Region In Exercises 5–16, find the area of the region.Interior of r = 4 – 4 cos θ

To determine

To Calculate:

The area of the interior of polar equation r=9sinθ that is above the polar axis.

Explanation

Given:

The provided polar equation r=9sinθ.

Calculation:

Shaded section above the polar axis of bounded by the curve r=9sinθ is shown below:

From the figure, it can be seen that polar region lies between 0 to 2π.

The region above the polar axis is traced out for 0θ2π.

Substitute r=9sinθ for r, 2π for β and 0 for α

It is known that the area of shaded region bounded by graph of r=f(θ) between the radial lines is θ=α and θ=β is:

A=12αβ[f(θ]2.dθ

Substitute these values in above formula and get,

A=1202π[9sinθ]2.dθ

A=1202π[81+sin2θ18sinθ]

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