Find the volume of the solid E bounded by z = 3+x2 +y2 and z = 6.
The Attempt at a Solution
I'm going to use cylindrical coordinates. So, I have,
z = 3 + r2
Clearly, my bounds on z are 3 and 6. If I project the intersection of the paraboloid and the plan onto the xy-plane, I have 3 = r^2. My bounds on r are then 0 and ±√3, but I reject the negative boundary. I will use the bounds 0 and 2*pi for θ, though I don't know how these bounds are known to be appropriate. So, if I perform a triple integral of r in the order dz dr dθ, I get an answer of 9*pi.
Does that sound legitimate?