Area Integral for unknown portion of circle.

My problem requires me to write an iterated integral for the area inside the circle r=2cos(theta) for -pi/2 <theta< pi/2 (circle of radius 1 centered at (1,0)) and outside the circle r=1 centered at the origin.

So I can write an iterated integral for the circle of radius 1 centered at (1,0) and then subtract the area portion of the circle at the origin. But what is this portion of area? Can I write another iterated integral for this area portion and then subtract it from the iterated integral? How would I approach this?

Any help would be greatly appreciated! Thanks!

 

So i was able to write an integral for the area of the circle, and tried integrating for the area between the curves of each circle. I am getting a value for the requested area (as described above) that seems unreasonable and do not know where I am making the mistake, or if I am evening doing it correctly. Any suggestions?

 

I think 2pi/3 would be the answer if the left side of the shape were bounded by straight lines as opposed to curved ones. Since this is an approximation to the area, I'll probably use it if I cannot figure something else out. Thanks so much for your help!

 

I think this integral may work for the Area outside the circle, r=1 and inside the circle, r=2cos(theta)

polar coordinates:

The double integral (where 1 <= r <= 2cos(theta) and -pi/3 <= theta <= pi/3 ) of differential Area dA=rdrdtheta

This seems to make sense when visualizing the double integral.

Solving, I got Area = 1.91 which seems reasonable since it is about sixty percent of a circle of radius 1 and area pi.

Does this seem like a valid method?