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(Solved): In this exercise, we characterize the area of the sphere of radius R in n dimensions, denoted S_(n)( ...
In this exercise, we characterize the area of the sphere of radius
Rin
ndimensions, denoted
S_(n)(R), and the volume of the spherical ball bounded by the spherical surface, denoted by
V_(n)(R). The spherical surface in
ndimensions is defined by
\Sigma _(n)={(vec(r))inR^(n)|x_(1)^(2)+x_(2)^(2)+dotsx_(n)^(2)=R^(2)}and the spherical ball (or solid sphere) is the domain bounded by the surface
\Sigma _(n):
RnV_(n)(R)=\int dots\int_(x_(1)^(2)+x_(2)^(2)+dotsx_(n)^(2)<=R^(2)) dx_(1)dx_(2)dotsdx_(n)Rn=10