<s>[INST] integrate $2\sqrt{1-x^2/4-y^2/9}$ I am trying to find the volume of the ellipsoid $E$ given by $x^2/4 + y^2/9 + z^2 \leq 1$ by computing $\iiint_E dV$
I ended up with $$\int_{-2}^{2}\int_{-3\sqrt{1-x^2/4}}^{3\sqrt{1-x^2/4}}\int_{-\sqrt{1-x^2/4-y^2/9}}^{\sqrt{1-x^2/4-y^2/9}}1dzdydx$$
After doing the inner integral I get $$\int_{-2}^{2}\int_{-3\sqrt{1-x^2/4}}^{3\sqrt{1-x^2/4}}2\sqrt{1-x^2/4-y^2/9}\space dydx$$
I am not sure about how to figure out the next one
The hint is to use trig substitution, though I am not sure if it is talking about this step or the next one
 [/INST] $$\int_{-2}^{2}\int_{-3\sqrt{1-x^2/4}}^{3\sqrt{1-x^2/4}}2\sqrt{1-x^2/4-y^2/9}\space dydx$$ is the same as $$\int_{-2}^{2}\int_{-\sqrt{9-9x^2/4}}^{\sqrt{9-9x^2/4}}2\sqrt{1-x^2/4-y^2/9}\space dydx$$ then do trig sub so $\frac{y^2}{9}=1-\frac{x^2}{4}\sin^2{\theta}$ 
then $y=\sqrt{9-\frac{9x^2}{4}}\sin{\theta}$, $dy=\sqrt{9-\frac{9x^2}{4}}\cos{\theta}d\theta$
then you get $$\int_{-2}^{2}\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}6(1-\frac{x^2}{4})\cos^2{\theta}d\theta dx=\int_{-2}^{2}\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}3(1-\frac{x^2}{4})(1+\cos{2\theta})d\theta dx$$
then you can keep going to end up with $8\pi$
(this was the given solution)
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