volume of a sphere

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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