Can someone shed a little bit of light on the problem of volumes of revolution about the $x$-axis and the $y$-axis of the same shapes? Take for example $f(x)=x^2$. If we want to find the volume bounded by the parabola and the $x$-axis with axis of revolution at $y=0$ we would use the standard method of disks and get the volume $\pi/5$ for $x=[0,1]$. Now if we want to find the volume by rotating the curve around the $y$-axis instead, using the cylindrical shells method, we get the volume to be $\pi/2$.
Now, maybe it's just my flawed intuition, but since we are basically rotating the same shape/area by 360$^\circ$ (but in different "directions"), I would guess the two volumes to be the same. Anyone knows some good graphics/animation/resources to help me visualize this problem?