- #1
rc3232
- 1
- 0
Homework Statement
Calculate volume of the solid region bounded by z = √(x^2 + Y^2) and the planes z = 1 and z =2
Volume in spherical coordinates
- Thread starter rc3232
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Homework Help Overview
The discussion revolves around calculating the volume of a solid region defined by the equation z = √(x² + y²) and bounded by the planes z = 1 and z = 2, within the context of spherical coordinates.
Discussion Character
- Exploratory, Assumption checking, Mathematical reasoning
Approaches and Questions Raised
- Participants explore the visualization of the solid and discuss integrating over specified bounds. Questions arise regarding the appropriate limits for integrals in spherical coordinates, and some suggest using cylindrical coordinates instead.
Discussion Status
Participants are actively discussing different methods to approach the problem, including the use of spherical and cylindrical coordinates. There is a recognition of the need to visualize the solid to better understand the volume, and some guidance has been offered regarding plotting and setting limits for integration.
Contextual Notes
There are mentions of specific limits for integration based on the geometry of the solid, as well as the suggestion to plot the region to clarify the volume's shape. The discussion reflects varying interpretations of the best coordinate system to use for the problem.
Calculate volume of the solid region bounded by z = √(x^2 + Y^2) and the planes z = 1 and z =2
- #2
dikmikkel
- 160
- 0
Edit: You could visualize it and integrate over 1 and add these volumes.
Last edited:
- #3
RC32
- 1
- 0
it's a cone, but how do you set the limits for the different integrals in spherical coordinates?
- #4
The Gringo
- 6
- 0
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]
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
- 837
- 4
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]
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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