Area of the shaded region

Question

What integral will give the area of the shaded region?

r=cos2θ

With half of one petal as the shaded region.

Answer 1

Area in Polar is 1/2 Integral of (cos (2x))^2 dx from 0 to pi/4. Using double angle formula we get

1/2 Integral of (cos(4x + 1))/2 dx which after calculation will be = 0,19635.

Answer 2

Hi Arthur:

when is r=0: cos(2*θ)=0 means 2θ=pi/2 so θ=pi/4

Next we can use A= ∫ (1/2)*r² dθ

r²= cos^2(2*θ)

so A=(1/2) ∫ cos^2(2*θ) dθ where the integral bounds are 0 and pi/4

remember cos^2(θ) = (1+cos(2*θ))/2 so that cos^2(2*θ) = (1+cos(4*θ))/2

This hint should give you enough to finish the problem.