We have to calculate the absolute value of
$$\left | \int_s \vec{F}. \vec{dA} \right |$$ of the vector field $F(x,y,z) :=(e^y,0,e^x) $ on the surface $$S:=\left \{ (x,y,z)|x^2+y^2=25, 0≤z≤2, x≥0, y≥0 \right \}$$
solution i tried-The given vector field is $F(x,y,z) :=(e^y,0,e^x) $ and we know form Gauss's Theorem that $$\int_s \vec{F}. \vec{dA}=\int_v \nabla.\vec{F}dv$$ now according to this if calculate the divergence of my given field vector it will be zero $$\vec{\nabla}.(e^y,0,e^x)=0$$ so the answer will be zero .But there is some non zero number in answer ,is there something that i'm missing ?
Please help
Thankyou