Application of Gauss' Divergence Theorem

Question

The problem is as follows:

My attempt towards solution:

What am I doing wrong here? What does the $(-1)$ signify on the right hand side? Thank you.

Answer 1

Using the divergence theorem.

If $S_1$ is your target. $S_2,S_3,S_4, S_5$ are the sides cut by the planes $x = 0, y=0, z = 0, z = 4,$ respectively.

Gauss' divergence theorem says:

$\iint_{S_1} A\cdot dS_1 + \iint_{S_2} A\cdot dS_2 + \iint_{S_3} A\cdot dS_3 + \iint_{S_4} A\cdot dS_4 + \iint_{S_5} A\cdot dS_5 = \iiint \nabla\cdot A\ dV$

You can calculate:

$\int_0^4\int_0^6 y\ dy\ dz + \int_0^4\int_0^3 2x\ dx\ dy + \int_0^3\int_0^{1-2x} 4\ dy\ dx -36$

or

$\int_0^4\int_0^3 2(6-2x) + 2x\ dx \ dz$

And both should give you the same answer.