What is the Correct Equation for Volume and Its Units in this Physics Problem?

[NewtonianAlch]

Homework Statement

In the equation dm = δ x 2∏rLdr

Where δ = density, and

2∏rL = volume

How is it that the volume can be 2∏rL? The units of r is (metres) and the units of L is (metres) which leads to m² (Area)

Should it not be ∏r²L for volume?

The Attempt at a Solution

My reasoning that it's 2∏rL is because dm and dr are infinitesimally small such that the circle formed doesn't really have a surface area as such and therefore the volume is simply the circumference multiplied by the length...but that still doesn't make it a "volume"

Can anyone explain?

Thanks

 

[jedishrfu]

Looks like you forgot the units of dr which is also meters.

The 2*pi*r*L is the area of a cylinder so 2*pi*r*l*dr is a volume element with dimensions of meters^3

 

[milesyoung]

You forgot to include the thickness dr in your calculation of volume. It has units of metres.

This way of setting up an integral by considering infinitesimally small quantities is something you'll find to be very common in a lot of physics texts. It's not mathematically rigorous, but it's often a useful heuristic.

If you want some intuition on how it works in a more mathematical setting, try looking up the definition of the integral as the limit of a Riemann sum.