Definite integral over a semicircular area $\int_0^{2a}\int_0^{\sqrt{2ax-x^2}}\frac{\phi'(y)(x^2+y^2)x}{\sqrt{4a^2x^2-(x^2+y^2)^2}}dy\,dx$. change the order of integration in
$$\int_0^{2a}\int_0^{\sqrt{2ax-x^2}}\frac{\phi'(y)(x^2+y^2)x}{\sqrt{4a^2x^2-(x^2+y^2)^2}}dy\,dx$$
I was able to change the order of integration here to 
$$\int_0^a\int_{a-\sqrt{a^2-y^2}}^{a+\sqrt{a^2-y^2}}\frac{\phi'(y)(x^2+y^2)x}{\sqrt{4a^2x^2-(x^2+y^2)^2}}dx\,dy$$
Now i am stuck with the integration here w.r.t $x$. I tried substituting $(x^2+y^2)^2$ with $t$ but then replacing value of $4a^2x^2$ becomes a problem. I was thinking of substituting both $x$ and $y$ as $k\cos\theta$ and $k\sin\theta$ but then $y$ won't be constant w.r.t $x$, so that too I believe is out of the question!

Next thing you want to do is re-scale
$$\int_0^a\int_{a-\sqrt{a^2-y^2}}^{a+\sqrt{a^2-y^2}}\frac{\phi'(y)(x^2+y^2)x}{\sqrt{4a^2x^2-(x^2+y^2)^2}}dx\,dy =$$ $$ a^3 \int_0^1 \phi'(a y) \,dy \int_{1-\sqrt{1-y^2}}^{1+\sqrt{1-y^2}}\frac{(x^2+y^2)x}{\sqrt{4 x^2-(x^2+y^2)^2}}dx$$
After this you can prove the second integral is constant. From there the solution is easy.
