Volume in spherical coordinates

  • Thread starter rc3232
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  • #1
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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

 

Answers and Replies

  • #2
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Edit: You could visualize it and integrate over 1 and add these volumes.
 
Last edited:
  • #3
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it's a cone, but how do you set the limits for the different integrals in spherical coordinates?
 
  • #4
In sphereical coordinates you know that [itex]x=\rho\cos\theta\sin\phi[/itex], [itex]y=\rho\sin\theta\sin\phi[/itex] and [itex]z=\rho\cos\phi[/itex]
You can use this to find limits for [itex]\rho[/itex].
If you draw the x-z or y-z plane intercept this can help you find [itex]\phi[/itex]
 
  • #5
DryRun
Gold Member
838
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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:
[tex]\int \int \int dr d\theta dz[/tex]
 

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