I need to solve $\displaystyle\iint\frac{x^2}{(8x^2+6y^2)^{\frac 3 2}}$ on the domain $8x^2+6y^2\leq 1$.
I recognise this is an improper integral, so we need a monotonic series of domains $D_n\rightarrow D$ on which we'll calculate the integral. Can I have $D_n=8x^2+y^2+1/n$?
The integral is difficult to solve. I guess there's an easier way to approach it.
Consider the transformation $$(x,y) = \left(\frac{r}{\sqrt{8}} \cos \theta, \frac{r}{\sqrt{6}} \sin \theta\right),$$ which results in a much simpler integrand. You will notice that the Jacobian $4 \sqrt{3} r$ will cancel out the singularity at $0$.