Spherical limits of integration for a region bounded by a cone and a praboloid

[s18]

Hi everybody, I am trying to solve the following problem and I get stuck on the last question. I would appreciate a lot that someone helps me .

Here is the problem: Let D be the region bounded from below by the cone z= the root of (x^2 + z^2), and from above by the paraboloid z = 2 – x^2 – y^2
1. Set up and evaluate the iterated integral in cylindrical coordinates that gives the volume of D using the order of integration dz dr dΘ
2. Set up ( without evaluating) the iterated integral in cylindrical coordinates that gives the volume of D using the order of integration dz dr dΘ
3. Set up ( without evaluating) the iterated integral in sphericalrical coordinates that gives the volume of D using the order of integration dρdФdΘ

For the 1st question, I get : z between r and 2 - r^2
r between 0 and sqrt2
the angle between 0 and 2 pi

For the second question, I find 2 integrals : for the first one : r between 0 and sqrt(2-z), z between 1 and 3, same angle.
For the integral two : r between 0 and z, z between 1 and 0, same angle.

However, for the third question, when I rty finding the limits of integration for ρ, i get the following equation : ρcosσ + ρ^2 - (ρcosσ)^2 - 2 = 0 and I cannot solve it.

Could someone help me. THANKS A LOT.

 

[algebrat]

Did you use intersection of cone and paraboloid to find limits? The cone should be of less importance (just tells you angles from z-axis). The radial limits should be from one point on the parabola to another point on the parabola. If you're having trouble visualizing, try integrating the region in the plane between y=x^2 and y=|x|, but using polar coordinates.