Can someone check if my solution to the following problem is correct.
(a) Evaluate
$$\int_{0}^{\alpha\sin\beta} \int_{y\cot\beta}^{\sqrt{{a}^{2}-{y}^{2}}} \log({x}^{2}+{y}^{2}) \,dx\,dy$$
(b) Change the order of integration in the integral in (a)
For part (a), I let $x=\alpha cos(\beta)$ and $y=\alpha sin(\beta)$, then applying the change of variable formula
$\int_{0}^{y} \int_{x}^{-x} \alpha\log({\alpha}^{2}) \,d\alpha\,d\beta$
$\int_{x}^{-x} \alpha\log({\alpha}^{2}) \,d\alpha\,d\beta=0$
So $\int_{0}^{y} \int_{x}^{-x} \alpha\log({\alpha}^{2}) \,d\alpha\,d\beta=0$
For part (b), the reverse order of integration becomes
$\int_{0}^{\alpha\cos\beta} \int_{\sqrt{{a}^{2}-{x}^{2}}}^{x\tan\beta} \alpha\log({\alpha}^{2}) \,d\beta\,d\alpha$
thank you in advance
So the region of integration is a sector of a disk $x^2+y^2\le a^2$ with angle $\beta$ at the center. So in polar coordinates the integration becomes $$\int_0^{\beta}\int_0^ar\log r^2 dr d\theta=\beta(\int_0^ar\log r^2 dr)=\beta(\frac{a^2}2(2\log a-1))$$
Change of variables is as follows $$\int_0^{a\cos \beta}\int_0^{x\tan\beta}\log(x^2+y^2)dydx+\int_{a\cos\beta}^a\int_0^{\sqrt{a^2-y^2}}\log(x^2+y^2)dydx$$