Calculate volume enclosed by cylinder and paraboloid (integration).

Look first at the $xy$-plane (the bottom). The condition limits the area $D$ between $y=x^2$ and $y=1$. It is bounded in $(x,y)$. Now look at what happens along the vertical $z$ axis. It says: take those points $(x,y,z)$ that are between $z=0$ and $z=x^2+y^2$. The set (and the volume) is finite, it is between two surfaces ($xy$ plane and the paraboloid).

Try to split integration as $$ \iint_D\int_{z=0}^{z=x^2+y^2}\,dz\,dxdy. $$

First of all the volume is not infinite because it is bounded at $z=2$.
Draw the figure on a $xy$-plane at $z=t$ where $0\leq t\leq2 $ you will see that the figure gets closed.
$\int_{0}^{2}S(t)dt$ where $S(t)$ is the area enclosed by $y=1,y=x^2$ and $x^2+y^2=t$.
Keep in mind that you will have to consider two cases separately when $0\leq t\leq1 $ and $1\leq t\leq2 $.