# Spherical Coordinates - Help me find my bounds

1. Apr 2, 2015

### RJLiberator

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

A vase is filled to the top with water of uniform density f = 1. The side profile of the barrel is given by the surface of revolution obtained by revolving the graph of g(z) = 2 + cos(z) over the z-axis, and bounded by 0 ≤ z ≤ π. Find the mass of the vase.

2. Relevant equations

3. The attempt at a solution

So I know that Mass = the triple integral of density dV.
I need to find my bounds.

I would like to use spherical coordinates as it seems to make sense with this problem.

Theta goes from 0 to 2pi.
phi goes from 0 to pi ? since it is bounded by 0 and pi? Or am I wrong here?
and rho goes from 0 to something involving the g(z) equation, but I am not sure how to manipulate it :/.

Any tips to get me started or confirmation on the phi bounds would be great.

2. Apr 2, 2015

### Dick

Cylindrical coordinates make much more sense since you have the radial limit as an explicit function of z.

3. Apr 2, 2015

### Svein

Therefore, in cylindrical coordinates:
Yes, and r=g(z).

4. Apr 2, 2015

### RJLiberator

Ah, so I assume the wrong coordinates. I guess this makes sense since a barrel is a cylinder... Sigh. 0_o. Let me try some work now.

5. Apr 2, 2015

### RJLiberator

So I am getting:

Theta: from 0 to 2pi.
r from 0 to 2+cos(z)
z from 0 to pi.

This would mean we integrate in the order dr dz dtheta

6. Apr 2, 2015

### RJLiberator

So I performed the triple integral with the bounds:
Theta: from 0 to 2pi.
r from 0 to 2+cos(z)
z from 0 to pi.

It seems to be right, my one concern is with the bounds on r. It's clear to me that it starts at 0, but why am I allowed to put 2+cos(z) as the upper limit. So z=r in cylindrical coordinates, and the function was g(z)=2+cos(z) so it was that simple?

7. Apr 2, 2015

### Dick

Sound right. I'm not sure what you are saying with 'z=r in cylindrical coordinates'. The function g(z) defines the largest value of r for a given value of z, isn't that what the problem description implies?

8. Apr 2, 2015

### RJLiberator

Hm. I see.

The function g(z) definitely defines the largest value for r, that makes more sense to me. r is the distance from the z-axis in cylindrical coordinates and that is what is implied by the question (bounds).

That makes sense to me. The other bounds are rather obvious, so that seems to work.