You'll need a triple integral, and in my opinion, it'd be easiest to do this using spherical coordinates. If any of this is confusing, let me know and I can simplify it using only rectangular coordinates.

You'll have a triple integral in the order of d(rho)d(phi)d(theta). The limits on theta will be 0 to 2pi (because it's a sphere), the limits on phi will be 0 to pi/2 (the angle it forms with the z axis), and the limits on rho will be 0 to 5 (the distance from the origin to the curve). In the integrand will be 1, since you're finding volume. Understand, or do I need to use rectangular coordinates?