Let $\mathcal{S}$ be the square in the $xy$-plane shown in the figure below, oriented with the normal pointing in the positive $z$-direction. Estimate $$\iint_{\mathcal{S}} \mathbf{F} \cdot \,d\mathbf{S}$$ where $\mathbf{F}$ is a vector field whose values at the labeled points are $$\begin{array}{llll} \mathbf{F}(A) &= \left<-6,1,-6\right>,&\qquad \mathbf{F}(B) &= \left<5,9,1\right>\\ \mathbf{F}(C) &= \left<-5,4,-2\right>,&\qquad \mathbf{F}(D) &= \left<2,-2,4\right>\end{array}$$

$\iint_{\mathcal{S}} \mathbf{F} \cdot \,d\mathbf{S} \approx$