I want to calculate the volume of $E = \{(x,y): x,y>0, (x+y)^3 <axy \}$ where $a>0$.
I defined, $U= (0, \infty) \times (0, \pi/2), V=\{(x,y): x,y>0\}$ And then $T:U \to V$ by $T(r, \theta) = (r\cos^2(\theta), r\sin^2(\theta))$.
It is easy to check that $T$ is diffeomorphism and $J_T(r, \theta) = r\sin(2\theta)$ where $J_T(r, \theta) = \det(D_T(r, \theta))$.
Also, $T^{-1}(E) = \{(r, \theta) \in U : r < a\cos^2(\theta)\sin^2(\theta)\}$ Therefore, using the well-known theorem about variable-change and also Fubini's theorem, we get the following:
\begin{align} V(E) & = \int_E 1\,dx\,dy = \int_{T^{-1}(E)} |J_T(r,\theta)|\,dr\,d\theta \\[8pt] & = \int_{T^{-1}(E)} r\sin(2\theta) \, dr \, d\theta= \int_0^{\pi/2}\left[ \int_0^{a\cos^2(\theta)\sin^2(\theta)} r \sin(2\theta)\,dr \right]\,d\theta \\[8pt] & = \int_0^{\pi/2} \frac{a^2}{2}\sin(2\theta)\cos^4(\theta)\sin^4(\theta)=\frac{a^2}{32} \int_0^{\pi/2}(\sin(2\theta))^5 \end{align}
I am not really sure if what I did here is correct. Is it ok? I am not sure because the integral left to calculate is not "nice", and in the question there was a hint to use the substitution $x=r\cos^2(\theta), y = r\sin^2(\theta)$ which is what I did here, so I am wondering if what I did here is correct as the integral left is more annoying than I expected, so maybe I did something wrong.
Thanks!