Recursive Partitioning of an Interval and a Geometric Interpretation of the Trapezoidal Rule Error

This paper proposes a geometric construction that allows one to represent the definite integral of a function of one variable as the sum of the areas of a trapezoid and an infinite series of triangles obtained by successively subdividing the original segment. It is shown that for sufficiently smooth functions, this series converges to the exact value of the integral. The connection with Romberg's method is discussed, and examples for quadratic functions are given.