Find the volume of the solid inside the sphere x^2 + y^2 + z^2 = 4 and over the paraboloid 3z = x^2 + y^2
The Attempt at a Solution
This should be easy to calculate using polar coordinates. The limits for z is [r^2/2, sqrt(4-r^2)] and for tetha: [0, 2*pi], but how do I find the limits for r? The intersection between the two surfaces is: sqrt(4 - r^2) = r^2/3. By inspection I can see that the answer is r = sqrt(3), but what is the mathematical method to find this limit?