Green's theorem says that: $$ \int_C L \ dx + \int_C M \ dy = \iint_D \frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} \ dx \ dy $$ If the M and N statisfy $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = 1$, then the formula can be used to compute the area of region D bounded by curve C. When $M=x, L=0$ or when $M=0, L=-x$ then the formula can be interpreted as approximating the area using rectangles and when $M=\frac{x}{2}, L=-\frac{y}{2}$ then it can be interpreted as approximating the area using triangles. There are infinitely many functions that satisfy $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = 1$. What does this condition geometrically mean? What is the geometrical interpretation of area formulas found using Green's theorem?
Edit: I came up with one interpretation of the formula, but I don't think it is very intuitive.
$\int \frac{\partial L}{\partial y} \ dy = L(x, y) + C_1$ and $\int \frac{\partial M}{\partial y} \ dx = M(x, y) + C_2$. Let $f(x, y) = \frac{\partial L}{\partial y}$ and $g(x, y) = \frac{\partial M}{\partial x}$. Then $\int f(x) \ dy = L(x, y) + C_1$ and $\int g(x) \ dx = M(x, y) + C_2$. If we think of f(x, y) and g(x, y) as surfaces then L(x, y) and M(x, y) represent the areas of slices of these surfaces. Then the condition $\frac{\partial M}{\partial x} - \frac{\partial L}{\partial y} = f(x) - g(x) = 1$ would mean that the height of the surface f(x) - g(x) is constant and equal to 1. The left hand side then calculates the volume of each surface separately by dividing it into slices and calculating the volume of each slice using the fundamental theorem of calculus. The sign in the left hand side is changed, because when intergating around a continous curve the x bounds go from a smaller value to a bigger value and y got from a bigger value to a smaller value or the other way around.