I need to calculate the volume enclosed by: $$x^2 + y^2 = z, \space y = x^2, \space z=0, \space y = 1$$
The shape of the volume I get gets me confused. It is a paraboloid ($x^2 + y^2 = z$) intersected with cylinder ($y = x^2$) and limited by specific $z$ and $y$ plains. When I tried drawing this I saw that the volume is not limited by the "upper" $z$ plain, therefore it seems to be infinite. Did the lecturer provide us with "wrong" conditions, so the volume is infinite?
Am I right? If yes, how can I calculate the volume if I change my previous condition ($z = 0, \space y = 1$) to $0\le z \le 1$? I tried approaching this "updated" problem, but also didn't have any luck.
Any help would be appreciated.
EDIT: The answer including the integral solution was posted - see below. The whole problem was caused by me thinking about the volume "inside" the paraboloid, while the task was to calculate it "outside", enclosed by the cylinder.