<s>[INST] how to find out the region between two surfaces Describe the region cut out of the ball $x^2+y^2+z^2\le4$ by the elliptic cylinder $2x^2+z^2=1$ i.e the region inside the cylinder and ball
I equated $4-x^2-y^2=1-2x^2$ getting $y^2-x^2=3$. I guessed the region has to be 
$\sqrt{1-2x^2}\le z \le \sqrt{4-x^2-y^2}$, $-\sqrt{y^2-3}\le x \le\sqrt{y^2-3},-\sqrt{3} \le x\le \sqrt{3} $.
I am not sure if I am on the right path. I don't understand how to get theses regions analytically since it is very difficult to see these regions graphically by hand which was the case for two dimensional objects.
Is there any fool proof method for seeing the regions??
 [/INST] If we want to describe the region in $1/4$ of all space, i.e; $$x\ge0, z\ge0$$ then it becomes:
$$\sqrt{1-2x^2}\le z \le \sqrt{4-x^2-y^2},~~ 0\leq x\leq\frac{\sqrt{2}}{2},~~-\sqrt{x^2+3}\le y \le\sqrt{x^2+3}\\ 0\le z \le \sqrt{4-x^2-y^2},~~ \frac{\sqrt{2}}{2}\leq x\leq 2,~~-\sqrt{x^2+3}\le y \le\sqrt{x^2+3}$$

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