Calculate the volume of a generalized n-dimensional sphere:
First, the surface of a sphere is the set of all points a given distance from the origin in n-space. So, by the distance formula, a sphere of radius r is the set of all points which satisfy the equation
x1sq+x2sq+...+xnsq=rsq
More conveniently for our purposes, we can parameterize the surface like so:
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Justification: