I take it that the region is rotating about the y-axis, right?
So effectively we have the curve $x=f(y)=y^2$ rotating in the interval $0\le y\le 4$. Therefore the volume is given by the integral
$ \int_0^4\pi f(y)^2\,dy=\int_0^4\pi y^4\,dy=\frac{1024\pi}{5}. $
Edit: Sorry, (new guy here) I should be extra careful in answering the question rather than just giving the answer to what might be a homework problem :-)
The way I think (and teach my students to think) about a problem like this is that whenever the curve is rotating about the $y$-axis instead of the more common $x$-axis, you should write the boundary curves in the form $x=f(y)$. In other words, try to find the inverse function. After all, the disk/washer method of computing volumes doesn't care, which axis is the axis of rotation. Or in other words, the shape and volume of the object does not change, if you switch the labels of the two axes!