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Each of the integral expressions that follow represents the area of a region in the plane bounded by a function expressed in polar coordinates. Use the ideas from this section and from Chapter 9 to sketch the regions, and then evaluate each integral

40π4COS22θdθ

Short Answer

Expert verified

The value of the integral isπ2square unit

Step by step solution

01

Step 1. Given information

Integral:

40π4COS22θdθ
02

 Step 2. Plot the curve 

Since the area of a function is r=f(θ)is12abr2dθ

So by comparing the given integral we get,

r=22cos2θ

So the graph of this curve is:

03

Step 3. Evaluate integral

40π4cos22θdθ=40π41+cos4θ2dθ=20π41+cos4θdθ=2θ+sin4θ40π4=2π4+sinπ4-0=2π4-0=π2

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