Understanding Area in polar coordinates

[ttiger2k7]

One of the problems I am working on is confusing me.

Given r=2cos(theta) and r=2sin(theta), you get a graph with two circles intersecting each other at pi/4.

So I worked it out, using the Area of a Polar Region formula.

$$\int\frac{1}{2}\pi*r^{2}$$. When I got an answer, I checked and saw that it was incorrect. I noticed in the solutions guide that when they had started, they doubled the integral before working it out. That is the part the confused me. Why would you want to double the integral in this case? The enclosed region is all contained in one quadrant, so wouldn't just integrating from 0 to pi/4 be sufficient?

I'd like some help understanding this.

 

[lippyka]

When you double the integral, you are adding up both circles. The formula for the area of a polar region is \int\frac{1}{2}\pi*r^{2}, but this only works for one circle. When two circles intersect each other, you need to add up both circles, so you double the integral. This is because the area of the enclosed region is the sum of the areas of both circles. So by doubling the integral, you are able to calculate the area of the entire region inside the two circles.