#int int_S ((2x^2y),( - y^2), (4xz^2)) * d bb S#
By the divergence theorem, flux for the entire surface is:
#= int int int_V \ 4xy - 2y + 8xz \ dV#
#= int_(x = 0)^2 dx\ int_(y = -3)^3 dy \ int_(z = - sqrt(9 - y^2))^( sqrt(9 - y^2)) dz qquad \ 2y(2x - 1) + 8xz #
#= int_(x = 0)^2 dx\ int_(y = -3)^3 dy \ qquad \ [2yz(2x - 1) + 4xz^2]_(- sqrt(9 - y^2))^( sqrt(9 - y^2)) #
#= int_(x = 0)^2 dx\ int_(y = -3)^3 dy \ qquad \ 4ysqrt(9 - y^2)(2x - 1) #
#= int_(x = 0)^2 dx\ qquad [ \ - 4/3 sqrt(9 - y^2)^(3/2)(2x - 1) ]_(y = -3)^3 = 0 #
That's the flux across the entire cylinder surface, including the end caps.
If they are meant to be excluded, then:
- End cap at #x = 0#, #bb hat n = langle -1 , 0 , 0 rangle#:
#int int_S ((2x^2y),( - y^2), (4xz^2)) * d bb S#
#= int int_S ((0),( - y^2), (0)) * ((-1),(0),(0)) \ dy \ dz = 0#
- End cap at #x = 2#, #bb hat n = langle 1 , 0 , 0 rangle#:
#int int_S ((2x^2y),( - y^2), (4xz^2)) * d bb S#
#= int int_S ((8y),( - y^2), (8z^2)) * ((1),(0),(0)) \ dy \ dz#
#= 8 int_0^3 \ dz int_(- sqrt(9 - z^2))^( sqrt(9 - z^2)) \ dy \ qquad y = 0#
