Put flat caps on each end of this tube, and then measure the volume of the resulting container.

Answer:

To plot the skin of the container, you plotted
       
where
    , , and .
To describe the whole container and everything inside it, you go with
              
with , , and .
Call the container and everything inside it and measure its volume by calculating  
       
       
       .
Now if Mathematica can hack this, you're out of here.

Haw.
       
in cubic units is measured by:

Done.