   Chapter 10.5, Problem 13E

Chapter
Section
Textbook Problem

# Finding the Area of a Polar Region In Exercises 5–16, find the area of the region.Interior of r = 5 + 2 sin θ

To determine

To calculate: The value of the area of the interior of polar equation r=6+5sinθ that is below the polar axis.

Explanation

Given:

The provided polar equation r=6+5sinθ

Calculation:

Shaded section below the polar axis of bounded by the curve r=6+5sinθ is shown below:

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

The section below the polar axis is traced out for πθ2π.

Substitute these values r=6+5sinθ for r, 2π for β and π for α

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

A=12αβ[f(θ]2.dθ

Substitute these values in the above formula, and get

A=12π2π[6+5sinθ]2.dθ

A=12π2π[36+25sin2θ+60sinθ]

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