# Homework Help: Solid of revolution question: given height of a cylindrical core in a sphere

1. Mar 1, 2010

### zeion

1. The problem statement, all variables and given/known data

A hole is drilled through the center of a ball of radius r, leaving a solid with a hollow cylindrical core of height h. Show that the volume of this solid is independent of the radius of the ball.

2. Relevant equations

$$V = \int_{a}^{b} 2\pi x (f(x) - g(x)) dx$$

$$V = \int_{a}^{b} \pi ([f(x)]^2 - [g(x)]^2) dx$$

3. The attempt at a solution

Am I allowed to use the radius of the cylinder? Then I can find the volume of the cylinder and then the radius of the sphere?

I can't really think of a way to do this without referring to any radius..

2. Mar 1, 2010

### Dick

The 'cylindrical core' isn't really a cylinder. It has rounded ends. Put the core around the x-axis and set up one of those integrals. The second one worked fairly well for me. Now just say what f(x) and g(x) are in terms of r and h.

3. Mar 1, 2010

### zeion

I thought I had to show that it was independent of the radius of the ball?

4. Mar 1, 2010

### Dick

You have to show the result of the integral V is independent of r. The formula for the integral V will contain r. You just have to show that if you do the integral then the result doesn't depend on r. f(x) and/or g(x) will depend on r. That doesn't necessarily mean V does.

Last edited: Mar 1, 2010
5. Mar 2, 2010

### holezch

hey, I am trying this too.. pretty interesting.

so I should do f(x) = volume of sphere and g(x) = volume of cylinder right ? how should I find the volume of the cylinder without knowing the radius of it... thanks, I need some more clues

6. Mar 2, 2010

### Dick

There is really no actual 'cylinder' here. There's a cylindrical cross section of a sphere. f(x) and g(x) aren't volumes. They are the radii of the disks or shells you want to deal with. Just set up the integral. It will magically work out. Trust me.

7. Mar 2, 2010

### holezch

oh..right.. I was thinking of taking off the spherical tops and then we will have a cylinder.

Okay, so (sorry, I'm new to this revolution stuff), we have g(x ) that will be shells for the cylindrical part and f(x ) can be shells for the overall sphere part? we would still need a radius for g(x), unless we can express it in a different way :S

edit : Okay, Ididn't know that radii meant radius in plural form.. so.. we have to set up f(x) and g(x) radii for the inner part and the outer part.. :S I have no idea where to start

thanks!

Last edited: Mar 2, 2010
8. Mar 2, 2010

### Dick

The length of the cylindrical cross section is h. The radius of the sphere is r. Find the radius of the cylindrical cross section by intersecting it with the sphere. Then find f(x) and g(x). Don't wimp out like zeion. You can probably use either shells or disks. I think I did disks. The integration is not even hard. Show the result is, in fact, independent of r.

9. Mar 2, 2010

### holezch

okay.. I couldn't find what "Find the radius of the cylindrical cross section by intersecting it with the sphere." is supposed to mean.. could anybody help me out..?
thanks

edit: okay, maybe I don't need to figure out what that was supposed to mean:
if I say that cr is the radius of the cylinder part.. then 2cr is the diameter, h is the length
of the hole, and then we can draw a triangle and get an expression for cr! now it should be easy enough with regular disk integration (I think)

Last edited: Mar 3, 2010
10. Mar 3, 2010

### holezch

okay, so now should I use the shell method and integrate from the radius of the cylinder to the radius of the sphere?

11. Mar 3, 2010

### Dick

That should work.

12. Mar 3, 2010

### holezch

thanks, so I'll something like:
r - radius of cylinder ( sqrt(4R^2 - h^2) )
$$2 \pi \int \sqrt{R^{2} - x^{2}$$
integral goes from r to R

I think I'll have to use a substitution x = Rcosu or something of the like, but I foresee a problem.. when I have to evaluate this integral from sqrt(4R^2 - h^2) to R^2.. it looks like this may get a little messy. Am I on the right track doing this substitution? thanks..

13. Mar 3, 2010

### Dick

sqrt(R^2-x^2) is half the length of your shell, right? Where's the radius of the shell? You integration expression is incomplete. And I don't quite agree with your expression for the radius of the cylinder either, I think you've got the diameter there.

14. Mar 3, 2010

### holezch

Right, I have to multiply whatever I get by 2 after. and you're right, I should divide that thing I got by 2, I had a diameter.

so it should be

$$4 \pi \int x \sqrt{R^{2} - x^{2}$$

from r to R , where r is $$\sqrt {4R^{2} - h^{2}}/2$$

Is that alright? do you think that I might have some trouble calculating this, since I'll have to evaluate at $$\sqrt {4R^{2} - h^{2}}/2$$ .. and should I use x = Rcosu?
thanks a lot

so, doing that:

$$4 \pi \int x \sqrt{R^{2} - x^{2}$$

let x = Rcosu
then
$$- \int R^{2}cosu \sqrt{R^{2}(1-cos^{2}u)} sinu du = - \int R^{3}sin^{2}(u)cos(u)du$$
, then
$$y = sinu, - R^{3} \int y^2 dy$$
the limits are all r to R, but for my last sub.. I have to change the limits to sin(r) and sin(R). Dealing with sin(r) and sin(R) is my problem.. should I not have done trigonometric substitutions?
thanks

Last edited: Mar 3, 2010
15. Mar 3, 2010

### Dick

You didn't need to do trig substitutions u=R^2-x^2 will do the job.

16. Mar 3, 2010

### holezch

thanks, I thought about that.. but I don't think that it's valid.. when you solve for x, the range of x of your new integrand is only positive, in our original integral, x can be negative or positive. I guess I'm wrong though? What have I mistaken?

17. Mar 3, 2010

### Dick

You are rotating the positive region around the y-axis. That's the full solid of revolution. I don't see any reason to worry about negative x values.

18. Mar 3, 2010

### holezch

okay, so I got it down to this with u = R^2 - x^2

$$-4 \pi \int \sqrt{u} du$$

after subbing and algebra. . at this point, I lost track of those constants :S

anyway, am I on the right track?

then

$$-4 \pi * 2(R^2 - x^2)^{3/2} / 3$$ evaluated at r to R

$$-4 \pi * 2(R^{2} - (4R^{2} - h^{2})/2)^{3/2} / 3$$

and I got lost here, am I right so far? thanks

Last edited: Mar 3, 2010
19. Mar 3, 2010

### Dick

You agreed that the limit (the radius of the cylinder) was sqrt(4R^2-h^2)/2, right? Where did the 2 go? You should check the other constants as well. They don't all look right. But the limit is the most important.

20. Mar 3, 2010

### holezch

right, I had a /2 in there, forgot to put it in with latex.. , and right, my constants were wrong, I had 2 * 2 * 2 and I put 6 anyway, I was more focused on getting rid of that R term, will it work out like this? I edited LATEX now, should that negative sign be there? I got it from du/-2x = dx
I can't get rid of that R^2 term when I expand that expression.. this is such a setback
thanks

Last edited: Mar 3, 2010
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