## How do you find the area of the region bounded by the polar curves r = cos ( 2 θ ) and r = sin ( 2 θ ) ?

### Question:

How do you find the area of the region bounded by the polar curves r = cos ( 2 θ ) and r = sin ( 2 θ ) ?

### Answer:

**The areas of both regions are π2.**

The graph of r=sin(2θ), 0≤θ<2π looks like this:

Since the area element in polar coordinates is rdrdθ, we can find the area of the four leaves above by

A=∫2π0∫sin(2θ)0rdrdθ.

**Let us evaluate the inside integral first,**

A=∫2π0[r2/2] sin(2θ)0dθ=∫2π0sin2(2θ)/2 x dθ

By the double-angle identity sin2(2θ)=1−cos(4θ)/2,

A=14∫2π0[1−cos(4θ)]dθ=1/4[θ−sin(4θ)/4]2π0=14[2π−sin(8π)/4−(0−sin(0)/4)]=1/4(2π)=π/2

**Hence, the area is π2.**

For r=cos(2θ), the area can be found by

A=∫2π0∫cos(2θ)0rdrdθ

## Answer ( 1 )

Answer Above.