Suppose that \(\mathcal{S}\) is a surface in \(\mathbf{R}^{3}\) with a parametrization \(\Phi\) whose domain \(\mathcal{D}\) is the square in Figure 14. The values of a function \(f\), a vector field \(\mathbf{F}\), and the normal vector \(\mathbf{N}=\mathbf{T}_{u} \times \mathbf{T}_{v}\) at \(\Phi(P)\) are given for the four sample points in \(\mathcal{D}\) in the following table. Estimate the surface integrals of \(f\) and \(\mathbf{F}\) over \(\mathcal{S}\).

Point Pin D A B C D 1 A B f 3 1 2 5 D FIGURE 14 F (2,6,4) (1, 1,7) (3,3,-3) (0, 1, 8) X N (1, 1, 1) (1, 1, 0) (1,0, -1) (2, 1,0)