Defining rho in spherical coordinates for strange shapes?

[Stirling Carter]

Homework Statement

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.

Homework 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.

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.

 

[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]

 

[Stirling Carter]

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.

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.

 

[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.

 

[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.

Thanks again for your time.

 

[Simon Bridge]

I do understand the concept of an integral.

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.

 

[Stirling Carter]

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.

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 ?

 

[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 $26^\circ$?)

 

[Stirling Carter]

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 $26^\circ$?)

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

 

[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?

 

[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 ?

 

[Simon Bridge]

That would be r = (cosθ-π/2).

... 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.

 

[Stirling Carter]

... 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.

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

 

[Simon Bridge]

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

 

[Ray Vickson]

Homework Statement

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.

Homework 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.

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.

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

 

[SteamKing]

Homework Statement

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.

Homework 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.

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