Finding integration bounds in cylindrical and spherical coordinates

[hytuoc]

How do I get the bounds for a function w/out drawing a graph??
Like, Volume of the solid bounded above by the sphere r^2+z^2=5 and below by the paraboloid r^2=4z. How would I get the bounds for these in cylindrical coordinate (r dz dr dtheta)?

***Mass of the solid inside the sphere p=b and outside the sphere p =a (a<b) if the density is proportional to the distance from the origin. How do I get the bounds for this problem in spherical coordinates (p^2 sin(phi) dp dphi dtheta)??
Pls show me how to get the bounds step by step...i really want to learn how to do this. Tahnks so much

 

[HallsofIvy]

The first problem is should be simple because the equations you have are already in cylindrical coordinates. The first thing you have to do, in any coordinates, if you want to integrate with respect to x and y after z, is project down to the xy-plane.
The parabola r²= 4z intersects the sphere r²+ z²= 5 where r²+ (r²/4)²= 5 or r⁴/16+ r²- 5= 0. That's the same as u²+ 16u- 80= (u- 4)(u+ 20)= 0. If u= 4, then r= 2 (Since u= r², we can't use the u= -20 solution.)
Because of the symmetry, θ (which doesn't appear in the formulas) ranges from 0 to 2π while r ranges from 0 (the middle) to 2. In the interior integral, z ranges from the paraboloid: z= r²/4 up to the sphere $z= \sqrt{5-r^2}$.

In the second problem, you have two concentric spheres with centers at the origin (I assume- you only mention ρ). φ and θ have no restrictions on them: their integrals will range from 0 to π (for φ) and from 0 to 2π (for θ). Of course, ρ will range from a to b.

 

[hytuoc]

Thanks much