Triple Integral Using Cylindrical Coordinates

[henryc09]

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.

Homework Equations

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]

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.

 

[henryc09]

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]

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.

Yes, your limits were correct.

They should lead you to an integral of rlog(2r-1) … you can use integration by parts on that.