- #1

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## Homework Statement

Use a triple integral in cylindrical coordinates to show that the volume of the solid bounded above by a sphere [tex]\rho = \rho_{o}[/tex], below by a cone [tex]\phi = \phi_{o}[/tex], and on the sides by [tex]\theta = \theta_{1}[/tex] and [tex]\theta = \theta_{2}[/tex], [tex]\theta_{1} < \theta_{2}[/tex] is

[tex] V = 1/3 \rho_{o}^3 (1-cos(\phi_{0}))(\theta_{2}-\theta_{1}) [/tex]

## Homework Equations

In cylindrical coordinates, the sphere has the equation [tex] r^2 + z^2 = \rho_{o}^2[/tex] and the cone has the equation [tex] z = r cot(\phi_{o})[/tex]. For simplicity, consider only the case [tex] 0 < \phi_{o} < \pi / 2 [/tex].

## The Attempt at a Solution

I had two methods, not sure which would actually work because I'm stuck.

[tex]

V = \int\int\int dV = \int\int\int r dr dz d\theta

[/tex]

[tex]

= \int\int\int\sqrt{\rho_{o}^2 - z^2} dr dz d\theta

[/tex]

[tex]

= \int\int\int\sqrt{\rho_{o}^2 - (r cot(\phi_{o}))^2} dr dz d\theta

[/tex]

Here, I don't know how I would change variables to integrate with respect to [tex]\phi[/tex] and [tex]\rho[/tex]. I could use the Jacobian, but I doubt that's right because the whole question itself could have been solved if I did that in the first place. It's easy to show that the Jacobian transform from cartesian to spherical is [tex]\rho^2 sin(\phi)[/tex], integrate that with respect to [tex]d\phi dr d\theta[/tex] with the given bounds and come up with

[tex] V = 1/3 \rho_{o}^3 (1-cos(\phi_{0}))(\theta_{2}-\theta_{1}) [/tex]. But that's not the answer because it doesn't go through cylindrical.

Attempt #2

[tex]V = \int\int z dA[/tex]

[tex]= \int\int \sqrt{\rho_o^2 - r^2} dA[/tex]

[tex]= \int\int \sqrt{\rho_o^2 - r^2} r dr d\theta[/tex]

And on this one, I think I might be able to do more if I put in the bounds of integration. I'm thinking [tex]-\sqrt{\rho_o^2-z^2} < r < \sqrt{\rho_o^2 - z^2}[/tex] but to get rid of [tex]z[/tex] that turns into [tex]-\sqrt{\rho_o^2-(r cot(\phi_{o}))^2} < r < \sqrt{\rho_o^2 - (r cot(\phi_{o}))^2}[/tex] which can't be used because it has a dependent limit on [tex]r[/tex], which is the variable I'm integrating with respect to.

Is there any guidance that someone can offer as to which approach, if any are even close, I should pursue, or any alternative suggestions? Thank you.