The volume, when y= y(x) is rotated around the x- axis is (by the disk method) .
In this case, that would be which gives the volume you quote.
But the moment is NOT just that volume times . It is, rather, that same integral with another x in the integral: . Use "integration by parts".
Since this is rotated around the x-axis, it is symmetric with respect to both y and z and it immediately follows that and .