No, you shouldn't. y^2+ z^2= x is a paraboloid with axis along the x-axis, not the y-axis. For this problem you should project onto the yz-plane.

Or, you can just "change" the problem- swap x and y and find the voume of the region bounded by x= 0, z= y, and z= x^2+ y^2. For that, you project onto the xy-plane.