VC5.32

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Using the cartesian form of Gauss divergence theorm,we have

\iint _{S}[(x+z)dydz+(y+z)dzdx+(x+y)dxdy]\,

=\iiint _{V}[{\frac  {\partial }{\partial x}}(x+z)+{\frac  {\partial }{\partial y}}(y+z)+{\frac  {\partial }{\partial z}}(x+y)]dV\,,V being the volume enclosed by S.

=\iiint _{V}(1+1+0)dV\,

=(1+1+0)dV\,

=2V\,

=2\times [{\frac  {4}{3}}\pi (2^{3})]\,

={\frac  {64\pi }{3}}\, since V is the volume of the sphere of radius 2.

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