Integration: Cylindrical to Spherical

[Knissp]

Homework Statement

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

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

Homework Equations

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

The Attempt at a Solution

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

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

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

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

Here, I don't know how I would change variables to integrate with respect to $$\phi$$ and $$\rho$$. 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 $$\rho^2 sin(\phi)$$, integrate that with respect to $$d\phi dr d\theta$$ with the given bounds and come up with
$$V = 1/3 \rho_{o}^3 (1-cos(\phi_{0}))(\theta_{2}-\theta_{1})$$. But that's not the answer because it doesn't go through cylindrical.

Attempt #2
$$V = \int\int z dA$$

$$= \int\int \sqrt{\rho_o^2 - r^2} dA$$


$$= \int\int \sqrt{\rho_o^2 - r^2} r dr d\theta$$

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

 

[Defennder]

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

This looks all right for the differential volume. But I think changing the order of integration would make it easier, otherwise you would have to integrate it piecewise for the spherical portion and the conical one. Try doing in the order $$rdzdrd\theta$$

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

This looks odd. Why did you have to express r in terms of z and phi_0? You could leave it as it is.

Here, I don't know how I would change variables to integrate with respect to $$\phi$$ and $$\rho$$. 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 $$\rho^2 sin(\phi)$$, integrate that with respect to $$d\phi dr d\theta$$ with the given bounds and come up with
$$V = 1/3 \rho_{o}^3 (1-cos(\phi_{0}))(\theta_{2}-\theta_{1})$$. But that's not the answer because it doesn't go through cylindrical.

You're supposed to do it in cylindrical coordinates right? And you don't have to do it in terms of spherical coordinates.

Attempt #2
$$V = \int\int z dA$$

I don't see how that gives you the required volume. The required volume doesn't have a cylindrical base.

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

Those limits don't make much sense. r is always positive.

 

[Knissp]

Try doing in the order $$rdzdrd\theta$$This looks odd. Why did you have to express r in terms of z and phi_0? You could leave it as it is.

You're supposed to do it in cylindrical coordinates right? And you don't have to do it in terms of spherical coordinates.

Thanks, I'm still a bit lost though. If I use $$r dz dr d\theta$$ as the order of integration like recommended, what are the limits for r? or do I change r to $$\rho sin(\phi)$$. And would z make sense from $$-\sqrt{\rho_o^2-r^2} < z < \sqrt{\rho_o^2 - r^2}$$?

 

[Defennder]

Thanks, I'm still a bit lost though. If I use $$r dz dr d\theta$$ as the order of integration like recommended, what are the limits for r? or do I change r to $$\rho sin(\phi)$$. And would z make sense from $$-\sqrt{\rho_o^2-r^2} < z < \sqrt{\rho_o^2 - r^2}$$?

You should figure out the limits from the innermost integrand to the outermost. So, you can imagine that an imaginary line from -infinity to +infinity along the z-axis would pass though the required volume. Where it enters and exits would define its limits. For r, it goes from zero to the cone, since the volume is limited by the confines of the cone.

Your limits for z is incorrect. As you have described them, those limits would define a volume of a sphere with radius p_0.

 

[Knissp]

Alright, now I have:

$$\int_{\theta_1}^{\theta_2}\int_0^{\rho_o sin(\phi_o)}\int_0^{\rho_o} r dz dr d\theta$$

$$= (\theta_2-\theta_1) \frac {\rho_o^3}{2} sin^2(\phi_o)$$

So that's not it...

 

[HallsofIvy]

Your limits of integration on z are wrong for several reasons. First, because the volume is bounded below by the cone, z does not go from 0 up to the cone. Second, $z= \rho_0 sin(\phi)$ only where the cone meets the sphere. You want z to go from the cone, for variable r, up to the sphere, again for variable r.

 

[Knissp]

$$\int_{\theta_1}^{\theta_2}\int_0^{\rho_o sin(\phi_o)}\int_{r cot(\phi_o)}^{\sqrt{\rho_o^2 - r^2}} r dz dr d\theta$$

I did it wrong still...

Here's what I was doing (correct me if I'm wrong, I know something has to be):
I think I see why 0 was not the lower bound for z. Rather, it should be any point on the lower part of the cone, because not all of these points are situated at zero. So looking at this cone, $$\phi = \phi_o$$, I made a triangle with legs z and r, and the angle opposite r is $$\phi_o$$. From this it follows that $$cot(\phi_o) = z / r$$ or $$z = r cot(\phi_o)$$, which is the lower bound for z.

For the upper bound, I had to use the sphere. It's equation can be written $$r^2 + z^2 = \rho_o^2$$, hence the upper limit for z: $$\sqrt{\rho_o^2 - r^2}$$.

 

[Defennder]

You sure you couldn't get the answer with those limits?

 

[Knissp]

With those limits, it comes out to be:

$$\frac {(\theta_2-\theta_1) (\rho_o^3) (((cos^3(\phi_o) - 1) + sin^2(\phi_o) cos(\phi_o))}{3}$$

 

[Defennder]

You're missing a negative sign somewhere. Furthermore, $$\cos^3 \phi_0 + \sin^2 \phi_0 \cos \phi_0 = \cos \phi_0 (\sin^2 \phi_0 + \cos^2 \phi_0)$$.

 

[Knissp]

I got it! Nice Identity! Didn't see that there! You guys are awesome, thanks!