Evaluating $\iiint_E(x^2+y^2+z^2)\ dx\ dy\ dz$ where $E$ is an ellipsoid

Question

I have a practice exercise involving triple integrals and I'm not really sure where to start.

Evaluate the triple integral $$\iiint_E(x^2+y^2+z^2)\ dx\ dy\ dz$$ where $E$ is defined by $(\frac xa)^2+(\frac yb)^2+(\frac zc)^2\le1$ with $a,b,c\in\Bbb R^+$.

Answer 1

HINT:First substitute $x=aX , y=bY , z=cZ$. Now use spherical coordinates

Answer 2

Adapt spherical coordinates $$\left.\eqalign{x&=r\cos\theta\cos\phi\cr y&=r\cos\theta\sin\phi\cr z&=r\sin\theta\cr}\right\},\qquad 0\leq r\leq R,\quad 0\leq\phi\leq2\pi, \quad -{\pi\over2}\leq\theta\leq{\pi\over2} $$ to the case of your ellipsoid, and write $$\left.\eqalign{x&=a\>t\cos\theta\cos\phi\cr y&=b\>t\cos\theta\sin\phi\cr z&=c\>t\sin\theta\cr}\right\},\qquad 0\leq t\leq 1,\quad 0\leq\phi\leq2\pi, \quad -{\pi\over2}\leq\theta\leq{\pi\over2}\ .\tag{1} $$ Your triple integral then becomes $$\eqalign{\int_E(x^2+y^2+z^2)\>{\rm d}(x,y,z)&=\cr\int_0^1\int_0^{2\pi}\int_{-\pi/2}^{\pi/2}&\bigl(x^2(\ldots)+y^2(\ldots)+z^2(\ldots)\bigr)\>J(t,\phi,\theta)\>d\theta\>d\phi\>dt\ ,\cr}$$ where the $\ldots$ indicate that you have to plug in the resulting expressions in the variables $t$, $\phi$, $\theta$ (e.g., $x^2=a^2t^2\cos^2\theta\cos^2\phi$), and $J$ denotes the Jacobian of $(1)$, which you have to compute beforehand.