I am confused about how to proceed with this exercise, which asks me to calculate the volume of $A = \left\{(x, y, z)\in\mathbb{R}^3; x^2+y^2 \leq z, z^2-2 \leq x^2+y^2\right\}$
I thought about polar coordinates, which give me
$$R^2 \leq z ~~~~~~~~~~~ R^2 \geq z^2-2$$
The range for $\theta$ is the same $[0, 2\pi]$ but how do I determine the range for $R$ or $z$?
$R^2 \leq z$ would imply $-z \leq R \leq z$ but it doesn't seem correct, also because the other condition $R^2 \geq z^2-2$ would give me a different range for $z$.
I need clarification on how to proceed and how to reason...
In cylindrical coordinates,
Paraboloid surface is given by $z = r^2$ and surface $z^2 - 2 = x^2 + y^2$ is given by $z^2 - 2 = r^2$
So at the intersection of both surfaces,
$2 + r^2 = r^4 \implies r = \sqrt2$
So limits of integration are,
$r^2 \leq z \leq \sqrt{r^2+2}, 0 \leq r \leq \sqrt2, 0 \leq \theta \leq 2\pi$