<s>[INST] How integrate $ \iint_{D} (\frac{x^2}{x^2+y^2})dA, \ \ \ \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0I'm trying to resolve this integral
$$
\iint_{D} (\frac{x^2}{x^2+y^2})dA,  \ \ \  \ D: x^2+y^2=a^2 \ \ and \ \ x^2+y^2=b^2, \ \ 0<a<b
$$
I tried with polar coordinates:
$$
x = r\cos{\theta} \\
y = r\sin{\theta} \\
Jacobian = r
$$
But I'm confused about in how to calculate the domain for each integral
 [/INST] if you are integrating over an annulus, then by symmetry
$$
\iint_{D} (\frac{x^2}{x^2+y^2})dA = \iint_{D} (\frac{y^2}{x^2+y^2})dA 
$$
so by adding you have 
$$
\iint_{D} (\frac{x^2+y^2}{x^2+y^2})dA = \iint_{D} dA = \pi(b^2-a^2)
$$
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