# Homework Help: Defining rho in spherical coordinates for strange shapes?

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1. Mar 28, 2016

### Stirling Carter

1. The problem statement, all variables and given/known data
The problem asks for a single triple integral (the integrand may be a sum but there must be a single definition for the bounds of the integral) representing the volume (in the first octant) of the shell defined by a sphere of radius 2 centered around the origin and a sphere of radius 1 centered around (0,0,1) in spherical coordinates. dV must be in the order dρdφdθ.

2. Relevant equations
For the large sphere, 4 = x^2 + y^2 + z^2, and for the smaller one 1 = x^2 + y^2 + (z+1)^2.

3. The attempt at a solution
I know the bounds for φ, which are 0 to π/2.
The bounds for θ are the also 0 to π/2.
The integrating factor is ρ^2sinφ, for dV dρdφdθ.
What I don't understand is how to define ρ. What is particularly confusing is that when 0 ≤ φ ≤ ≈ 26° there is no single radius that extends from the origin to the edge of the shell. You can't draw a straight line from origin to outer surface without going through the smaller sphere.

Surprisingly, I was unable to find a similar problem posted here or anywhere else. If you know of one, I would greatly appreciate that link. Thank you very much for any input you have to offer.

2. Mar 28, 2016

### Simon Bridge

Using $\rho$ for radial distance?
The outer bound depends on the outer sphere so you are fine there ... the inner bound depends on the angles, work out what that dependence is.
It may help to draw a sketch - perhaps a projection to a particular plane. Also to consider the volume as a difference between two volumes.

[slightly annoying that this is easier in cylindrical coordinates]

3. Mar 28, 2016

### Stirling Carter

Thanks for your reply. Drawing a projection onto the yz-plane did help me visualize the problem; what remains confusing is the bounds of ρ for 0 ≤ φ ≤ ≈ 26°. In these cases, is ρ considered to be the sum of the distance from the origin to the small sphere and the distance from the other side of the small sphere to the surface of the larger sphere? I don't understand how to integrate with respect to ρ when ρ is broken up in such a way.

4. Mar 28, 2016

### Simon Bridge

You are adding up the small volumes that are part of the shell ... you can think of is at adding the volumes that arev in the big sphere, then subtracting the volumes inside the little one... or adding the ones from the little sphere to the big one.

Check terminology: in spherical coordinates, it is usual for $\theta$ to indicate the angle from the z axis.
https://en.wikipedia.org/wiki/Spherical_coordinate_system

BTW: best practise - do all angles in radians.

Last edited: Mar 28, 2016
5. Mar 28, 2016

### Stirling Carter

I do understand the concept of an integral, I just don't understand how you can adequately describe what seems to need 4 bounds with 2 bounds. For example, at ρ extends from 0 to the lower edge of the small sphere. Then, ρ extends from the upper edge of the small sphere to 2. Here, I cannot say ρ goes from 0 to 2. I cannot say that it goes from the small sphere to the large sphere either, because a one segment would be left out. This is the source of my confusion.

6. Mar 28, 2016

### Simon Bridge

That is a problem - the integral is a summation. Revise.

For your problem - you have $$\int_0^{\pi/2 } \int_0^{\pi/2 } \int_{a(\theta ,\phi)}^{b(\theta ,\phi)} r^2\sin\theta \; dr d\theta d\phi$$ ... you are looking for hoe r depends on the angles. You actually have enough information to figure that out.

7. Mar 28, 2016

### Stirling Carter

Thanks for the note about the convention of what θ indicates. I'll keep that in mind in the future but I'll stick with what I've been doing right now for the sake of consistency.

I tried to define the smaller sphere in terms of ρ, φ, and θ:

x^2 + y^2 + (z+1)^2 = 1
x^2 + y^2 + z^2 + 2z + 1 = 1
x^2 + y^2 + z^2 + 2z = 0
ρ + 2z = 0
ρ + 2ρcosφ = 0
ρ(1 + 2cosφ) = 0
ρ = 0

Is this inconclusive because for the same φ and θ there are two different ρ's ?

8. Mar 28, 2016

### Simon Bridge

OK - so you are using $\theta$ for the angle from the x axis and $\phi$ for the angle from the z axis?
(This is what I used to do - for consistency with cylindrical coords.)

For a fixed $\phi$ how does $\rho$, for a point on the surface of the small sphere, vary with $\theta$ ?
(What is special about $26^\circ$?)

9. Mar 28, 2016

### Stirling Carter

Wouldn't ρ be the same for every fixed φ regardless of θ?

10. Mar 28, 2016

### Simon Bridge

Well done ... so you just need to know how $\rho$ varies with $\phi$ ... so what is the polar equation of a circle that is not centered at the origin?

11. Mar 28, 2016

### Stirling Carter

That would be r = (cosθ-π/2).
So I guess you could say that ρ is equal to (sinφ-π/2)
So the integral would be as follows:

∫0→π/2 ∫0→π/2 ∫sinθ-π/2→π/2 f(ρ,φ,θ) ρ^2 sinφ dρdφdθ ?

12. Mar 28, 2016

### Simon Bridge

... well, check: when $\theta = 0$, $\cos\theta -\pi/2 = 1-\pi/2$ but the actual circle needs to be 2 or 0... same for when $\theta = \pi/2$... I'd start from the top and work down, say $r=2\cos\theta$ (check) so when $\theta = \frac{\pi}{2}$, $r=0$.

Urge you to learn LaTeX.

13. Mar 28, 2016

### Stirling Carter

Thanks for all of your help! I will learn LaTeX before posting here again.

14. Mar 28, 2016

### Simon Bridge

No worries.
There is a PF LaTeX tutorial around here someplace; you only need the maths part.

15. Mar 28, 2016

### Ray Vickson

The surface $x^2+y^2+(z+1)^2 = 1$ is a sphere centered at (0,0 -1).

16. Mar 28, 2016

### SteamKing

Staff Emeritus
If the smaller sphere is centered at (0,0,1), shouldn't its equation be x2 + y2 + (z - 1)2 = 1 ?