Let S be the square in the xy-plane, oriented with the normal pointing in the positive z-direction. Estimate ∬S F⋅dS where F is a vector field whose values at the labeled points are F(A)=⟨9,3,5⟩, F(C)=⟨9,8,−4⟩, F(B)=⟨−9,8,−7⟩, F(D)=⟨−3,4,−1⟩.

Answer 1

Usually, this just the limit of the sum
∑F⋅n ∆S
but you have only provided a few discrete values of F, and have not indicated how S is to be subdivided.

Maybe you can review surface integrals and provide a bit more detail...

Answer 2

I disliked the above answer solely because this is a WebWork problem that requires an input, and "not enough information" is not a valid answer...