Use the divergence theorem to calculate the surface integral

Use the divergence theorem to calculate the surface integral ( two integration signs then letter S) F .dS for F(x,y,z)=x^2 z^3 i + 2xyz^3 j + xz^4 k , where S is the surface of the box with vertices

( + - 3 , + -2 , + - 1 )

I got a wrong answer.

Thank you so much !

Solution

grad F = 2x z^3 + 2x z^3 + 4x z^3 = 8x z^3

Hence, surface integral = volume integral over 8x z^3

Since z^3 is positive when z is positive and negative when z is negative, and zaries from +1 to -1, the volume integral will be zero.