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Hyper-volume by integration - Xorter - 04/08/2017

Hi, everyone!
My dream is to get a formula to get the n-dimensional hyper-volume of an n-dimensional function in cartesian AND polar coordinates, too! So the length of f(x), the area of f(x,y), the volume of f(x,y,z) ... etc. According to the other existing formulas I have created an own in cartesian coordinate system:

1st question: Do you find it correct?
2nd: How could it look in polar coordinate system?
(My final goal is to use these formulas to determine a few things about the base units of the hyperdimensional and interdimensional spaces from its derivatives and its existences. But for it, I need these formulas!)