V = int(S) z dxdy, S : x^2+y^2+z^2 = R^2
by applying Gauβ's divergence theorem,
int(S) z dxdy = int(V) ∂z(z)dV = int(V)dV, dV = dxdydz, V : x^2+y^2+z^2 <= R^2
= 4πR^3/3
if you solve the probrem directly without using Gauβ' divergence theorem,
V = int(S) z dxdy
= int(S) z^2/R dS
= int_0^2π dφ int_0^π R^3 sinθ cos^2θ dθ
= 2πR^3 int_0^π sinθ cos^2θ dθ
= 4πR^3/3
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