the question asks:
Use the Divergence Theorem to calculate the surface integral $ \iint \mathbf{F} \cdot d\mathbf{S}$; that is, calculate the flux of $\mathbf{F}$ across S. $\mathbf{F}(x,y,z)=e^xsiny\mathbf{i}+e^x cosy\mathbf{j}+yz^2\mathbf{k}$, S is the surface of the box bounded by the planes x = 0, x=1,y=0,y=1,z=0,and z=2.
my working as follows:
$\iint \mathbf{F} \cdot d\mathbf{S} = \iiint\limits_E div\mathbf{F} \cdot d\mathbf{V} = \iiint\limits_E (e^x-e^xsin(y)+2yz) d\mathbf{V}$
for the region $E= \{0\le x\le1, 0\le y\le1,0\le z\le2\}$
$\int_{0}^{2} \!\! \int_{0}^{1} \!\! \int_{0}^{1} \! (e^x-e^xsin(y)+2yz) d\ x \, d\ y \, d\ z = 2+2(e-1)cos(1)$
the correct answer is 2, could you please point out where I made the mistake? Thanks in advance!