Calculate $$I= \iiint\limits_{\Omega}2\,{\rm d}x\,{\rm d}y\,{\rm d}z$$ with $$\Omega: \left\{\begin{matrix} \left ( \frac{x}{3} \right )^{2}+ \left ( \frac{y}{2} \right )^{2}+ z^{2} & \leq & 1\\ z & \geq & 0 \end{matrix}\right.$$
My solution is:
Let $u= x/3, v= y/2, w=z.$ Using Jacobi's formula with: $$\left | J \right |= \begin{vmatrix} {x_{u}}' & {x_{v}}' & {x_{w}}'\\ {y_{u}}' & {y_{v}}' & {y_{w}}'\\ {z_{u}}' & {z_{v}}' & {z_{w}}' \end{vmatrix}= \begin{vmatrix} 3 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 1 \end{vmatrix}= 6\Rightarrow I= \iiint\limits_{\Omega}2\cdot 6{\rm d}u{\rm d}v{\rm d}w= 12\cdot \frac{1}{2}V= 6\frac{4\pi r^{3}}{3}= 8\pi$$
But seems like this way's not useful if I replace $x^{2}/9$ by $x.$ So I want to see another way to solve this problem without Jacobi's formula. Maybe I can understand when we should use Jacobi's formula. I used these to prep for my test. I need to the help, thank you.
Your $\Omega$ is half of the ellipsoid $E$ with semiaxes $3$, $2$, $1$. It follows that $$\int_\Omega 2\>{\rm dvol}={\rm vol}(E)=3\cdot2\cdot1\cdot{4\pi\over3}=8\pi\ .$$