Surface Integral Homework: ∫∫σ3x2 + 3y2 + 3z2 dS

[Baumer8993]

Homework Statement

Evaluate ∫∫_σ3x² + 3y² + 3z² dS
where σ is the part of the cylinder x² + y² = 4 between the planes z = 0
, and z = 1, together with the top, and bottom disks.

Homework Equations

Surface integrals, maybe divergence theorem?

The Attempt at a Solution

I am having trouble knowing where to start with this one. I think I need to do a surface integral, but maybe with more than one surface? If that is right then what would I do for the cylinder side? How would I handle the z in the integral?

 

[SteamKing]

What is the value of z at the top and bottom of the cylinder?

What is the equation of the surface in between the top and bottom disks? Hint: it's a constant

 

[haruspex]

I would start the integration over the curved surface by converting to cylindrical coordinates.

 

[Baumer8993]

Ok so I see that z = 0, and z = 1. What about the sides? Do I have to do them in three separate integrals?

 

[haruspex]

Ok so I see that z = 0, and z = 1. What about the sides? Do I have to do them in three separate integrals?

Yes.