Suppose I want to compute the integral
$$ \iiint_E f(x,y,z)\ dV $$
where $E$ is given by
$$ \frac{(x-1)^2}{4^2} + y^2+z^2\leq 1 $$
and
$$ f(x,y,z) = e^{xy+z} $$
I am thinking that I want to use spherical coordinates but I'm a little uncertain if my reasoning is correct about how to apply that to a shifted ellipsoid. I believe that I should use
$$ (x-1)/2 = \rho \sin\varphi\cos\theta $$
$$ y = \rho \sin\varphi\sin\theta $$
$$ z = \rho\cos\varphi $$
with
$$ 0\leq \rho\leq 1 $$
$$ 0\leq \varphi \leq \pi $$
$$ 0\leq \theta < 2\pi $$
and then treat this like a change of coordinates, so that I therefore compute the Jacobian and perform the change of variables in the integral.
But I'm not sure that my change of coordinates faithfully preserves the shape, or that it's the right one, or that there isn't a simpler way to do this. Typically when parameterizing any ellipsoid I have only parameterized the surface and therefore I can specify a scalar associated with the length along each axis. But here since I need a parameter $\rho$ I'm a little less confident in this approach, and I'm also a little concerned over whether the shift needs some other sort of treatment.
Best Answer
Traditionally with ellipsoids and triple integrals, some form of transformation of the form $(x,y,z) \to (au,bv,cw)$ is done first. If you're given an ellipsoid of the form:
$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} + \frac{(z-\ell)^2}{c^2}$$
Then a transformation would first look something like:
$$x-h = au, y-k = bv, z-\ell = cw$$.
A bit of manipulation shows that:
$$x = au+h, y = bv + k, z = cw + \ell$$
The Jacobian matrix of THIS is $abc$. Put in the proper values to turn your integral into $u,v,w$, and then now you're ready to turn this into the unit sphere, in other words:
$$ u = \rho \sin\varphi\cos\theta $$
$$ v = \rho \sin\varphi\sin\theta $$
$$ w = \rho\cos\varphi $$
This is the same way as you did it, but you've done it as one transformation instead of two. This should hopefully alleviate your concern that the transformation somehow isn't proper. You've done it correctly, but there's a little bit of extra machinery in the background that you've actually done in your head. You've gone straight from the ellipsoid to a spherical integral, but REALLY what you should be doing is going from the ellipsoid to the sphere, then using regular spherical coordinates.
My point is that both the methods are the same. But the method I'm using will hopefully outline a little bit better some of the things that are happening behind the scenes. Rather than going straight from Ellipsoid -> Spherical integral, you should go Ellipsoid -> Sphere -> Spherical integral.
I would even argue my way is easier, because you don't have to calculate all the nasty partial derivatives and do the determinant, and simplifying everything out. You can just use that the first jacobian is $abc$, then when you transform into regular spherical, you get a jacobian of $\rho^2\sin\varphi$ as expected. Let me know if you have any questions. Just a different way of explaining what's going on.