Find the mass of a piece of a cylinder

[Feodalherren]

Homework Statement

Find the mass of the piece of a cylinder x^2 + y^2 =1 that lies in the first octant above z=0 and below z=1-x.
The density is D=x.

Homework Equations

The Attempt at a Solution

I set up this integral:

\int^{\pi/2}_{0} \int^{1}_{0} \int^{1-rcos\theta}_{0} r^{2}cos\theta dz dr d\theta

Which ends up coming out as \frac{1}{3} - \frac{\pi}{16}

The correct answer is 1- \frac{\pi}{4}

 

[Feodalherren]

Nevermind I figured it out. They only wanted the "shell" mass. I had to do a surface integral.

 

[LCKurtz]

Homework Statement

Find the mass of the piece of a cylinder x^2 + y^2 =1 that lies in the first octant above z=0 and below z=1-x.
The density is D=x.

Homework Equations

The Attempt at a Solution

I set up this integral:

\int^{\pi/2}_{0} \int^{1}_{0} \int^{1-rcos\theta}_{0} r^{2}cos\theta dz dr d\theta

Which ends up coming out as \frac{1}{3} - \frac{\pi}{16}

The correct answer is 1- \frac{\pi}{4}

Do you recall the discussion in your previous post about the mass of a funnel? There was confusion whether your problem was to give the mass of a solid or a surface. You have the same issue in this problem. You have worked it as the mass of a solid piece of a cylinder in the first octant under that plane. Are you sure you aren't supposed to be thinking of a thin metal sheet forming just the curved surface of that cylinder? In that case you would want a surface integral. It would help if you would state the problem completely and exactly as it appears in your book. My guess it that it is supposed to be a surface integral.

[Edit] I see you figured it out while I was typing my response.

 

[Feodalherren]

This question actually came off an old quiz and it was stated on it exactly as I typed it. It seems counter intuitive to me anyway. Isn't the surface just infinitely thin and thus without mass? This whole idea seems weird to me. How could a surface possibly have mass?

 

[vela]

It can be a useful approximation for some physical situations when one dimension is negligible in comparison to the other two.

 

[Feodalherren]

How do you mean? Could you give me an example of a physical interpretation? One that's applicable to engineering would be awesome :).

 

[vela]

Things that are thin, like a sheet of paper, a steel plate, gold foil, etc.

 

[Feodalherren]

Hmm yeah ok fair enough. Thanks.