area of the region that lies inside both curves

**dammitpoo** · September 5th 2008, 11:39 PM

Hi,

I am trying to solve this problem:

Find the area of the region that lies inside both curves.

$ r_{1}=\sin\theta, r_{2}=\cos\theta $

Which are the circles $x^2+(y+\frac{1}{2})^2=\frac{1}{4}$ and $(x+\frac{1}{2})^2+y^2=\frac{1}{4}$ on the Cartesian coordinate system.

When I try to solve the problem in the following way, I get a different answer than in my textbook:

$ A_{1}=\int_0^\frac{\pi}{4}\frac{1}{2}\sin^2\theta\ d\theta=\frac{1}{4}\int_0^\frac{\pi}{4}(1-\cos2\theta)\ d\theta=\frac{1}{4}(\frac{\pi}{4}-\frac{1}{2})=\frac{\pi}{16}-\frac{1}{8} $
$ A_{2}=\int_0^\frac{\pi}{4}\frac{1}{2}\cos^2\theta\ d\theta=\frac{1}{4}\int_0^\frac{\pi}{4}(1+\cos2\th eta)\ d\theta=\frac{1}{4}(\frac{\pi}{4}+\frac{1}{2})=\fr ac{\pi}{16}+\frac{1}{8} $
$ A=A_{1}+A_{2}=(\frac{\pi}{16}-\frac{1}{8})+(\frac{\pi}{16}+\frac{1}{8})=\frac{\p i}{8} $

However, if I do it by symmetry, it gives me a different answer (the correct answer in my textbook):

$ A=2\int_0^\frac{\pi}{4}\frac{1}{2}\sin^2\theta\ d\theta=\frac{1}{2}\int_0^\frac{\pi}{4}(1-\cos2\theta)\ d\theta=\frac{1}{2}(\frac{\pi}{4}-\frac{1}{4})=\frac{\pi}{8}-\frac{1}{4} $

Can anyone explain this to me? I've been trying to figure it out, but I can't seem to get it.

**Laurent** · September 6th 2008, 01:58 AM

I think for $A_2$ you would need to integrate from $\frac{\pi}{4}$ to $\frac{\pi}{2}$ .

Laurent.

**dammitpoo** · September 6th 2008, 02:20 AM

I see it now. Thanks. =)

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