Let's say I'm trying to find the volume inside $x^4+y^4+z^4=1$. Can I go about defining new variables $X=x^2,Y=y^2,Z=z^2$ so that we know have a unit sphere in $X,Y,Z$. Then computing the Jacobian of this inverse transformation to be $(2x)(2y)(2z)=8xyz$. So that the Jacobian corresponding to this transformation is $\frac{1}{8xyz}$ and then I need to compute:
$$\int_{0}^{2\pi} \int_{0}^{\pi} \int_{0}^{1} \frac{1}{|8\rho^3 \sin (\phi)^2 \cos(\theta) \sin(\theta) \cos (\phi)|} \rho^2 \sin(\phi)\ \mathrm d\rho\ \mathrm d\phi\ \mathrm d\theta$$
Would that work?