Figure 4

Single limit at infinity n → ∞ of double sums of triangularly divided areas.

The area surrounded by f(x, y) = 0 and $\overline{AB}$ is$\begin{array}{cc}\hfill \sigma & =\frac{1}{2}\underset{n\to \infty }{lim}\sum _{k=1}^n\sum _{j=1}^k\left(\Delta x_k\Delta y_j-\Delta y_k\Delta x_j\right)\hfill \\ =\frac{1}{2}{\iint }_{f\left(x,y\right)\le 0}^{\left[A,B\right]}\left(dxdy\prime -dydx\prime \right).\hfill \end{array}$

The area of the triangle OAB is $\frac{1}{2}\left(x_{\text{B}}{y}_{\text{A}}-y_{\text{B}}{x}_{\text{A}}\right)$.