   Chapter 10.4, Problem 10E

Chapter
Section
Textbook Problem

# Sketch the curve and find the area that it encloses.10. r = 1 − sin θ

To determine

To Find: The area of the region that the polar equation r=1sinθ encloses.

Explanation

Given:

The polar equation is r=1sinθ.

Calculate the area of the region using the expression.

A=12abr2dθ=1202π(1sinθ)2dθ=1202π(12+sin2θ2sinθ)dθ

Substitute 1cos2θ2 for sin2θ.

A=1202π(12+1cos2θ22sinθ)dθ=1202π(1+12cos2θ22sinθ)dθ=1202π(32cos2θ22sinθ)dθ

Integrate with respect to θ and get the expression below

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