How to setup an integral in spherical coordinates for the volume of p = 2 sin O(theta

In summary, the conversation discusses finding the volume of a region enclosed by a spherical coordinate surface, with the use of spherical coordinates for the limits of the integral. There is some confusion about the upper limits of the integral and a correction is made. The resulting graph resembles a donut without the hole centered at the origin.
  • #1
VinnyCee
489
0
Here is the problem:

Find the volume of the region enclosed by the spherical coordinate surface [tex]\rho = 2 \sin\theta[/tex], using spherical coodinates for the limits of the integral.

Here is what I have:

I don't know if this is right, but here it is [tex]\int_{0}^{2\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{2\sin\theta}\;\rho^2\;\sin\theta\;d\rho\;d\phi\;d\theta[/tex]
 
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  • #2
It looks okay.Did u plot it?Heh,u we use [itex] r,\varphi,\vartheta [/itex] as the spherical coordinates...:wink:

Daniel.
 
  • #3
Whoops!

I posted the wrong thing. Instead of [tex]2\sin\theta[/tex] for the first integrals upper limit, it should be [tex]2\sin\phi[/tex]?

[tex]\int_{0}^{2\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\int_{0}^{2\sin\phi}\;\rho^2\;\sin\theta\;d\rho\;d\phi\;d\theta[/tex]

So confusing!

Is that wrong now?

The graph looks like a donut without the hole centered at the origin. I will post it shortly.
 
  • #4
No,it looks okay.

Daniel.
 

1. What are the basic steps for setting up an integral in spherical coordinates?

The basic steps for setting up an integral in spherical coordinates are:
1. Identify the region of integration and sketch it.
2. Determine the limits of integration for each variable (r, θ, and φ).
3. Convert the function to spherical coordinates.
4. Rewrite the function in terms of r, θ, and φ.
5. Set up the triple integral using the appropriate limits of integration and integrand.

2. How do I determine the limits of integration for r, θ, and φ in spherical coordinates?

The limits of integration for r, θ, and φ can be determined by examining the region of integration and considering the values of each variable that correspond to the boundaries of the region. For example, if the region is a sphere with radius a, the limits for r would be 0 to a. The limits for θ would be 0 to π for a full revolution around the z-axis, and the limits for φ would be 0 to 2π for a full revolution around the x-axis.

3. How do I convert a function to spherical coordinates?

To convert a function to spherical coordinates, you will need to use the following conversions:
x = r sinθ cosφ
y = r sinθ sinφ
z = r cosθ
Once you have substituted these values into the function, you can simplify and rewrite it in terms of r, θ, and φ.

4. Can I use spherical coordinates for any type of integration?

No, spherical coordinates are only suitable for integration when the region of integration is spherical or has spherical symmetry. In other cases, it may be more appropriate to use other coordinate systems such as Cartesian or cylindrical coordinates.

5. How can I check if my setup for an integral in spherical coordinates is correct?

To check if your setup for an integral in spherical coordinates is correct, you can try evaluating the integral using a calculator or computer software. If the result matches your expected answer, then your setup is likely correct. You can also double-check your limits of integration and converted function to ensure they are accurate.

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