Volume between sphere and outside cylinder.

1. Dec 25, 2012

peripatein

Hi,
1. The problem statement, all variables and given/known data
I am trying to compute the volume bounded by the sphere x2+y2+z2 = 4 and outside the cylinder x2+y2=2x.

2. Relevant equations

3. The attempt at a solution
The cylinder's equation is obviously (x-1)2+y2=1, but I am not sure how to formulate the integration for the volume. Using cylindrical coordinates, should the formulation be: ∫(θ=0,2π)dθ∫(r=1,2)dr∫(z=0,sqrt(4-r2))?

2. Dec 25, 2012

Lucas Mayr

have you tried to find the volume of the sphere and the cylinder and then subtracting?
making two integrals instead of one, might be easier to see, also use wolframalpha to plot the graph, might give you some hint as to what the ranges are.
also you can use spherical coordinates for one and cylindrical on the other

Last edited: Dec 25, 2012
3. Dec 25, 2012

haruspex

That still leaves the bulk of the difficulty - finding the volume of the cylinder that lies inside the sphere.
Using cylindrical or spherical coordinates here looks messy. How about sticking to Cartesian and taking slices orthogonal to the x axis? Each slice through the cylinder will be a rectangle with segment-of-circle endcaps.

4. Dec 25, 2012

Lucas Mayr

you could try to separate the two halves, upper and lower,and multiply by 2 at the end, let's say we take the upper half of the cylinder, then,
z goes from 0 to R*Cos∂ (the angule from Z to R),
if you cand find ρ, then 0<θ<2Pi and 0<∂<Pi/2
and you have the volume for the upper part of the cylinder

5. Dec 25, 2012

haruspex

I think you're overlooking the complexity of the cylinder's shape. What does the line of intersection of cylinder and sphere look like?

6. Dec 26, 2012

peripatein

Haruspex, supposing I follow your advice and stick to Cartesian - should I find solid of revolution of cylinder around y-axis? If so, wouldn't it be pi*int(0,2) [x^2*dx*dy/dx], which gives pi*int(0,2) [(x^2-x^3)/sqrt(2x-x^2)]?

7. Dec 26, 2012

Dick

I don't think cylindrical coordinates look at all bad. x^2+y^2=2x has a pretty simple form in cylindrical coordinates. What is it?

8. Dec 26, 2012

peripatein

r^2=2rcos(theta), hence r=2cos(theta)?

9. Dec 26, 2012

Dick

Right. So that's a circle. What range of theta do you need to cover the circle? Think about graphing it.

10. Dec 26, 2012

peripatein

0 to 2pi?

11. Dec 26, 2012

Dick

Graph it. It's really best if you keep r >=0 everywhere. Figure out which points on the circle correspond to which values of theta.

Last edited: Dec 26, 2012
12. Dec 26, 2012

peripatein

Is it not a circle whose center is at (0,0) and whose radius is 2?

13. Dec 26, 2012

Dick

No. It's center is at (1,0) and radius 1. Look back at what you got in the problem statement. Which values of theta correspond to which points on the circle?

14. Dec 26, 2012

peripatein

Theta = 0 corresponds to (2,0), theta = pi/2 corresponds to (1,1) etc.? Is that what you meant?

15. Dec 26, 2012

Dick

That's exactly what I mean, but pi/4 corresponds to (1,1), not pi/2. What point corresponds to pi/2?

16. Dec 26, 2012

peripatein

Could you please first explain why pi/4 corresponds to (1,1)?

17. Dec 26, 2012

Dick

r=2*cos(pi/4)=2*sqrt(2)/2=sqrt(2). So r=sqrt(2), theta=pi/4. That's the point (1,1).

18. Dec 26, 2012

peripatein

Okay, I understand why. But I still don't understand how r=2cos(theta) corresponds to a circle whose center is at (1,0) and radius is 1. Would you care to explain, please?

19. Dec 26, 2012

Dick

You wrote that the equation is (x-1)^2+y^2=1 in cartesian coordinates. Doesn't that tell you what it is? The point to figuring out which points on the circle correspond to which values of theta is to figure out a range of theta that will cover the circle once to use as a range in the integration.

20. Dec 26, 2012

peripatein

I understand all that, but my question was rather slightly different (I think). Supposing I didn't have that Cartesian equation, and only had r=2cos(theta), how could I have figured out where the center of the circle was, and its radius? Here's a guess: should I have simply needed to find the Cartesian equation first, before attempting to properly describe the circle? I mean, could I have derived C(1,0) and r=1 without having any knowledge of the Cartesian formulation?

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