Volume of ice cream cone triple integral

[Panphobia]

Homework Statement

Find the triple integral for the volume between a hemisphere centred at $z=1$ and cone with angle $\alpha$.

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.

 

[mfb]

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.

 

[Panphobia]

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.

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

 

[mfb]

Probably.

 

[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.

 

[mfb]

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

 

[LCKurtz]

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

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