As the trapezoidal rule for integration finds the area under the line connecting the endpoints of a panel, Simpson's rule finds the area under the parabola which passes through 3 points (the endpoints and the midpoint) on a curve. In essence, the rule approximates the curve by a series of parabolic arcs and the area under the parabolas is approximately the area under the curve. There is a unique curve with the equation

y = ax² + bx + c

passing through the points (-x,y₀), (0,y₁), and (x,y₂). There is a unique solution for a, b, and c generated by the three equations:

y₀ = a(-x)² + b(-x) + c

y₁ = c

y₂ = a(x)² + b(x) + c

The area under the curve from -x to x is

but the part in the square brackets can be rewritten as y₀ + 4y₁ + y₂ and so

For the adjoining parabola, y₂ is a collocation point; it is evaluated twice. The number of collocation points is one less than the number of parabolas. The series of coefficients for the y_i's for N points then is