How to transform triple integral $\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$

Question

I have stumbled across this triple integral $$\iiint_\Omega \sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz$$ where $$\Omega =\left\{(x,y,z)\in{\cal{R}}^3\ \bigg| \ \frac{x^2}{a^2}+ \frac{y^2}{b^2} + \frac{z^2}{c^2} \le1 \right\} $$

I know that I am supposed to use integral transform in which I have to use substitutions. I would substitute $x/a = u, y/b = v, z/c = w$. I can’t find enough examples on how to solve integrals with 3 variables with integral transform. I really struggled with this integral. Can you guys help me with it?

Answer 1

Make the variable changes $x= a u$, $ y= b v$, $z= c w$, and then integrate in spherical coordinates

\begin{align} &\iiint_{\frac{x^2}{a^2}+ \frac{y^2}{b^2} + \frac{z^2}{c^2}\le1 }\sqrt{1- \frac{x^2}{a^2}- \frac{y^2}{b^2} - \frac{z^2}{c^2} }\ dx dy dz \\ =&\ abc \iiint_{u^2 +v^2+w^2\le1 }\sqrt{1- {u^2}- {v^2} - {w^2}}\ dudv dw \\ = &\ 2\pi \ abc\int_0^\pi \int _0^1 \sqrt{1- {\rho^2}}\ \rho^2 \sin\phi \ d\rho d\phi =\frac{\pi^2}4 abc \end{align}