<s>[INST] Volume of ellipsoid outside sphere I have the ellipsoid $\frac{x^2}{49} + y^2 + z^2 = 1$ and I want to calculate the sum of the volume of the parts of my ellipsoid that is outside of the sphere $x^2+y^2+z^2=1$
How to do this? I know the volume of my sphere, $\frac{4\pi}{3}$, and that I probably should set up some double- or triple integral and transform the coordinates to spherical coordinates and evaluate but I have to admit I'm stuck on how to set this up. 
 [/INST] Hints. 
1) If $(x,y,z)$ satisfies $x^2+y^2+z^2\leq 1$ then
$$\frac{x^2}{49} + y^2 + z^2 \leq x^2+y^2+z^2\leq 1.$$
What does this inequality mean?
2) $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}=1$ is the equation of an ellipsoid centered at the origin of semi-principal axes of length $a$, $b$, $c$, and its volume is $\frac{4\pi (a\cdot b \cdot c)}{3}$.
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