1. The problem statement, all variables and given/known data Find the volume of the solid inside the surfaces x^2+y^2+z^2=1 and z=x^2+y^2 2. Relevant equations With x=0: z=y^2 ; z=√(1-y^2) or y as a function of z then gives: y=√z ; y=√(z^2-1) 3. The attempt at a solution First, I attempted to sketch a cross section of the region by setting x=0 (y=0 would give a similar cross section), and graphing z=y^2and z=√(1-y^2) on the y-z plane. This gives me a picture of a cross section of the region with Area=∫√(1-y^2)-y[^2]... The problem is that I won't have a calculator to solve this type of problem and I can't figure out how to solve for the limits of integration... I set the z's equal and have: y^4+y^2-1=0 but I don't know how to solve that for y by hand, and I'd have a similar problem if I solved for y and set the functions of z equal. Because the solid is symmetrical about the z-axis, I know I could also solve for the volume by using V=∫∏(radius)^2.dz where radius is y(z)≥0, or distance from the z axis to the nearest of the two positive z(y) equations [from z=0 to z=1], but wouldn't I still need to solve for the intersection point of the two curves where y>0 to split the volume into two halves? If I could solve for the intersection in terms of z, call it z0, I could solve V=∫[0,z0] ∏(+√(z))^2.dz + ∫[z0,1] ∏(+√(z^2-1))^2.dz .... but again, I don't know what z0 would be, or how to find it without a calculator. Apologies for not using Latex, I don't really know how to work it yet. Edit: Or is there a way to convert to polar and solve?