# Triple Integral Using Cylindrical Coordinates

## Homework Statement

A conical container with radius 1, height 2 and with its base centred on the ground
at the origin contains food. The density of the food at any given point is given by
D(r) = a/(z + 1) where a is a constant and z is the height above the base.
Using cylindrical polar coordinates, calculate the total mass of food in the container.

## The Attempt at a Solution

ok so mass is the integral D(r)dV, and in cylindrical coordinates dV is rdrd$$\theta$$dz

I thought that you could probably do:
$$\int^1_0 \,dr$$$$\int^{2\pi}_0 \,d\theta$$$$\int^{2r-2}_0 \,dz$$

$$(ra/(z+1))$$

But this makes the integral very difficult and I don't think it's right. I'm pretty sure there's something wrong with my limits on dz. Any help would be appreciated

tiny-tim
Homework Helper
Hi henryc09! I'm pretty sure there's something wrong with my limits on dz. Any help would be appreciated

Yes, you've integrated from z = 0 to z = 2r - 2 …

why?? z doesn't depend on r, the limits of z are the same for all r. OK so how do I work out the limits of z? It can't just be from 0-2 because that would make it a cylinder? Still a bit confused.

tiny-tim
Homework Helper
Hi henryc09! (just got up :zzz: …)
I thought that you could probably do:
$$\int^1_0 \,dr$$$$\int^{2\pi}_0 \,d\theta$$$$\int^{2r-2}_0 \,dz$$

$$(ra/(z+1))$$

But this makes the integral very difficult and I don't think it's right. I'm pretty sure there's something wrong with my limits on dz. Any help would be appreciated
OK so how do I work out the limits of z? It can't just be from 0-2 because that would make it a cylinder? Still a bit confused.

I'm sorry … somehow I read it as a cylinder. 