Conceptual Problem (Integration)

[anonymousphys]

1. Why is 2(pie)rh multiplied by the dr (change in r) equal to the change in volume?
This is in the case of deriving the moment of inertia of a cylinder; the equation is (change in mass)=2(pie)(radius)(height)(change in radius)(density).

Homework Equations

The Attempt at a Solution

I don't quite understand how 2(pie)(radius)(volume)(change in radius) equals change in volume? Is there a proof for this? I can imagine this working mathematically but not conceptually.2(pie)(radius)(height) is just the surface area for each layer, but how does the change in radius come in? I know it has something to do with accounting for the thickness of each hollow cylinder?

All replies are much appreciated. thanks.

 

[Cryxic]

1. Why is 2(pie)rh multiplied by the dr (change in r) equal to the change in volume?
This is in the case of deriving the moment of inertia of a cylinder; the equation is (change in mass)=2(pie)(radius)(height)(change in radius)(density).

Homework Equations

The Attempt at a Solution

I don't quite understand how 2(pie)(radius)(volume)(change in radius) equals change in volume? Is there a proof for this? I can imagine this working mathematically but not conceptually.2(pie)(radius)(height) is just the surface area for each layer, but how does the change in radius come in? I know it has something to do with accounting for the thickness of each hollow cylinder?

All replies are much appreciated. thanks.

Well let's start off simple. We know that 2*pi*r is the circumference of a circle. So the circumference of a circle multiplied by a height h means that we now the have the surface area of a cylinder (excluding tops). Do you see that at least? Once we have the surface area, it's just one more dimension to get to volume. So if you multiply that surface area by an infinitesimal distance element dr, you get an infinitesimal volume element = dV = dA*dr. Integrate over all that and you get the total volume.

 

[tiny-tim]

Hi anonymousphys!

(have a pi: π )

I don't quite understand how 2(pie)(radius)(volume)(change in radius) equals change in volume? … how does the change in radius come in? I know it has something to do with accounting for the thickness of each hollow cylinder?

I suspect you're confused because you're calling dr the "change in" radius.

When you're integrating, "d" just means "a small amount of" …

change doesn't come into it.

You're dividing the big cylinder into a lot of tiny cylindrical shells, each of thickness dr.

Each shell has surface area 2πrh, so multiply that by the thickness (dr) to get the volume dv = 2πrh(dr), and by density to get dm = 2πρrh(dr).

(Then integrate to get ∫ 2πρrh dr)

In all these cases, "d" simply means a small amount.