   Chapter 10, Problem 100RE

Chapter
Section
Textbook Problem

# Finding the Area of a Polar Region In Exercises 97-100, find the area of the region.Interior of r   =   5 ( 1   −   s i n   θ )

To determine

To calculate: The area of the interior region of r=5(1sinθ)

Explanation

Given: The polar equation is r=5(1sinθ).

Formula used:

The identity:

sin2θ=1cos2θ2

Calculation: Given, r=5(1sinθ)

The graph of the polar equation r=5(1sinθ) is obtained as below:

Here, the curve is symmetrical about the initial line. Therefore, the required area is twice the area above the initial line. The limits of integration are θ=0 to θ=π.

The area is:

A=2[120πr2dθ]

That is,

A=2[120π(5(1sinθ))2dθ]=250π(12sinθ+sin2θ)dθ=250πdθ500πsinθdθ+0πsin2θdθ

Use the identity: sin2θ=1cos2θ2

Thus,

A=250πdθ50

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