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Each of the integrals or integral expressions in Exercise represents the area of a region in the plane. Use polar coordinates to sketch the region and evaluate the expressions.

20π/20sin3θrdrdθ

Short Answer

Expert verified

The integral's value is20π/20sin3θrdrdθ=π4

Step by step solution

01

given information 

Let consider the given integral is20π/20sin3θrdrdθ

02

Finding establish the expression

The goal of this issue is to sketch the region and assess the expression using polar coordinates 20π/20sin3θrdrdθ

Using, r=0,r=sinθandθ=0,θ=π/2

θ
r=sin3θ
00
π/6
1.0
π/4
0.7071
π/3
0
π/2
-1.0
03

Calculations 

20π/20sin3θrdrdθ=20π/2r220sin3θ=20π/2sin23θ02θ=20π/2(1cos6θ)4

Integrate in relation to θ,

20π/20sin3θrdrdθ=2{θ(sin6θ6}40π/2cosxdx=sinx

Pointing the limits,

20π/20sin3θrdrdθ=2{π/2(sin3π)60}4=π4

As a result, the integral value is20π/20sin3θrdrdθ=π4

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