- #1
rc3232
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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
Volume in spherical coordinates
- Thread starter rc3232
- Start date
In summary, To calculate the volume of the solid region bounded by z=√(x^2 + Y^2) and the planes z=1 and z=2, use cylindrical coordinates and integrate over the given limits. The region can be visualized as a cone and the volume can be found by evaluating the triple integral with the given limits.
Calculate volume of the solid region bounded by z = √(x^2 + Y^2) and the planes z = 1 and z =2
- #2
dikmikkel
- 168
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Edit: You could visualize it and integrate over 1 and add these volumes.
Last edited:
- #3
RC32
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it's a cone, but how do you set the limits for the different integrals in spherical coordinates?
- #4
The Gringo
- 6
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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
- 838
- 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]
Attachments
FAQ: Volume in spherical coordinates
1. What is the formula for calculating volume in spherical coordinates?
The formula for calculating volume in spherical coordinates is V = ∫∫∫ r^2 sin(θ) dr dθ dφ, where r is the radius, θ is the polar angle, and φ is the azimuthal angle.
2. How do you convert from Cartesian coordinates to spherical coordinates?
To convert from Cartesian coordinates (x, y, z) to spherical coordinates (r, θ, φ), use the following formulas:
r = √(x^2 + y^2 + z^2)
θ = cos^-1(z/r)
φ = tan^-1(y/x)
3. What is the range of values for θ and φ in spherical coordinates?
The polar angle θ ranges from 0 to π, while the azimuthal angle φ ranges from 0 to 2π.
4. How does volume in spherical coordinates differ from volume in Cartesian coordinates?
In spherical coordinates, volume is calculated using a triple integral and involves the use of the radius and two angles. In Cartesian coordinates, volume is calculated using a single integral and involves the use of three coordinates (x, y, z).
5. What are some real-life applications of volume in spherical coordinates?
Volume in spherical coordinates is commonly used in physics and engineering to calculate the volume of objects with spherical symmetry, such as planets, stars, and particles. It is also used in calculus to solve problems involving three-dimensional shapes with spherical symmetry.