Volume in spherical coordinates

[rc3232]

Homework Statement

Calculate volume of the solid region bounded by z = √(x^2 + Y^2) and the planes z = 1 and z =2

Homework Equations

The Attempt at a Solution

 

[dikmikkel]

Edit: You could visualize it and integrate over 1 and add these volumes.

 

[RC32]

it's a cone, but how do you set the limits for the different integrals in spherical coordinates?

 

[The Gringo]

In sphereical coordinates you know that x=\rho\cos\theta\sin\phi, y=\rho\sin\theta\sin\phi and z=\rho\cos\phi
You can use this to find limits for \rho.
If you draw the x-z or y-z plane intercept this can help you find \phi

 

[DryRun]

You should first plot it to know what the volume looks like.

The volume between z=1 and z=2 is that of a circular disk. You need to use cylindrical coordinates.

Description of the region:
For r and θ fixed, z varies from z=1 to z=2
For θ fixed, r varies from r=1 to r=√2
θ varies from θ=0 to θ=2∏

Plug the limits into the triple integral and evaluate to find the required volume:
\int \int \int dr d\theta dz