# Volume of ice cream cone triple integral

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1. Jan 25, 2016

### Panphobia

1. The problem statement, all variables and given/known data
Find the triple integral for the volume between a hemisphere centred at $z=1$ and cone with angle $\alpha$.

3. The attempt at a solution

What I tried to do first was to get the radius of the hemisphere in terms of the angle $\alpha$. In this case the radius is $\tan \alpha$. I already figured out this integral in cylindrical polar coordinates and Cartesian coordinates. I am having a lot of trouble with spherical coordinates. I am trying to get $\rho$ for the hemisphere by drawing the projection of the shape on the xz plane and trying to get a formula for a radial ray that hits the hemisphere. The formula for a circle that is shifted up by one is $x^2+(z-1)^2=\tan^2 \alpha$, this is the part where I am stuck trying to find the equation for $\rho$ in terms of the angle $\phi$ of the radial ray. The limits for $\theta$ and $\phi$ are really easy, but again $\rho$ I just can't seem to get.

2. Jan 25, 2016

### Staff: Mentor

Do you need the integral in spherical coordinates? It is messy, and the other coordinate systems are much easier.

You are missing y2 in your hemisphere formula.
You can find z as function of the two angles, plug it into the equation for the hemisphere (in the right place) and solve for ρ.

3. Jan 25, 2016

### Panphobia

So I will have to use the quadratic formula to solve for $\rho$?

4. Jan 25, 2016

### Staff: Mentor

Probably.

5. Jan 25, 2016

### LCKurtz

I disagree with mfb on this problem. In fact spherical coordinates is the natural and easiest way to work this problem. You didn't state it in your problem but I assume you are talking about the upper half of the sphere of radius $1$ centered at $(0,0,1)$. Its equation is $x^2+y^2+(z-1)^2 = 1$ or, expanding it, $x^2 + y^2 + z^2 = 2z$. If you change that to spherical coordinates you should get $\rho = 2\cos\phi$. So the $\rho,~\theta,~\phi$ limits are all very easy.

6. Jan 25, 2016

### Staff: Mentor

How could you fix the radius of the half-sphere to 1?

7. Jan 25, 2016

### LCKurtz

I couldn't. I scanned it too quickly and assumed it was the "standard" problem you normally see in calculus books.

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