Hypercylindrical Volume

I'm a Physics major and my research mentor has given me a math assignment. He described what he wanted but, since I don't have any training in this area of math, I don't really know where to turn. I am supposed to find the volume of a shape that wraps around onto itself in n dimensions. I think of it as a cylinder. Anybody know where I can find information about this?
 
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Use the integration techniques from multivariable calc.
 
Given the vaugeness of the question, I would be tempted to just integrate the volume parameterised by

[tex]S(r,\theta,z_1,z_2,\ldots,z_n) = (r\cos{\theta},r\sin{\theta},z_1,z_2,\ldots,z_n)[/tex]

But I think the question you might have been asked was to find the volume of a hypersphere. That's easily googled. And has a few interesting properties as you move through the dimensions
 
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robphy

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One important property of volume is that it is additive.
Partition your object into pieces whose volumes are easier to calculate... then add them up. That it wraps around onto itself shouldn't be a problem as long as you don't double count any volumes. (The lateral area of a cylinder ("the soup label") is the same as the area of the associated rectangle.)
 

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