   Chapter 10.5, Problem 9E

Chapter
Section
Textbook Problem

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

To determine

To calculate: The area of the region of one petal of the polar curve r=sin2θ.

Explanation

Given:

The polar curve is, r=sin2θ.

Formula used:

The area for a continuous polar curve r=f(θ) between the two radial lines θ=α and θ=β which is non-negative on the interval [α,β] where 0<βα<2π is,

A=12αβ[f(θ)]2dθ=12αβr2dθ

The power reducing identity, cos2θ=12sin2θ.

Calculation:

Consider the equation for the polar curve, r=sin2θ

Recall the area for the polar curve, A=12αβr2dθ.

Construct the table for the values of r and θ after that plot them on graph to evaluate the limits.

For θ=0, substitute 0 for θ.

r=sin2(0)=0

For θ=π6, substitute π6 for θ.

r=sin2(π6)=sin(π3)=32

For θ=π3, substitute π3 for θ.

r=sin2(π3)=sin(2π3)=32

For θ=π2, substitute π2 for θ.

r=sin2(π2)=sin(π)=0

For θ=π, substitute π for θ.

r=sin2(π)=sin(2π)=0

Construct the table, using various values of θ and r, for the function r=6sinθ

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