# Mass of air in cylinder column

• ~Sam~

## Homework Statement

The density of air in the lower atmosphere decreases exponentially with height: ρ = ρ0e−z/H
where ρ0 = 1.3 kg/m3 and H = 10 km. What is the mass of air in a cylindrical column of cross-sectional area 1 m2 and height z, as a function of z? How much mass is contained in such a column 1.0 km high?

## Homework Equations

Volume of a Cylinder= pi*r*h

## The Attempt at a Solution

I think I need to integrate from 0 to z to get the formula for mass. However, I'm not sure what the equation is that I need to integrate. I was thinking it might be something like intg[0,z] pi*r2e-z/Hρ0dz. Still if it that was, I see an issue with integrating e-z/H. I haven't gone to the second part because I need to know the first.

Looks good to me.

$$m=\int^{z}_{0}dm=\int^{z}_{0}\pi r^{2}e^{-z/H}\rho_{0}dz$$

of course a mathematician will tell you that integrating for 0 to z is bad notation but its unlikely to give you the wrong answer.

How would you integrate e-z/h?

By substitution for the argument of the exponential.

By substitution for the argument of the exponential.

I'm not quite sure what you mean. Do you integration by substitution? Could you elaborate or give an example?

Yes integration by substitution. Its a pretty elementary integration, look in a calculus book. Sorry I don't have time right now to explain.