An integration-via-summation formula

In this paper by A.A. Ruffa, this formula is shown to derive from the method of exhaustion of the ancient Greeks which was used to find "the area of a shape by inscribing inside it a sequence of polygons whose areas converge to the area of the containing shape".

Accordingly the author call it the generalized metthod of exhaustion. Two proofs of the formula are given and the case of an infinite boundary for the integral is discussed. Several decompositions of elementary functions are also given.

Hint:

For $n=2$ $$\frac{-f(\frac12)}2+\frac{-f(\frac14)+f(\frac24)-f(\frac34)}4=- \frac{f(\frac14)+f(\frac24)+f(\frac34)}4$$ and $n=3$ $$\frac{-f(\frac12)}2+\frac{-f(\frac14)+f(\frac24)-f(\frac34)}4+\frac{-f(\frac18)+f(\frac28)-f(\frac38)+f(\frac48)-f(\frac58)+f(\frac68)-f(\frac78)}8\\= -\frac{f(\frac18)+f(\frac28)+f(\frac38)+f(\frac48)+f(\frac58)+f(\frac68)+f(\frac78)}8. $$

You see the pattern. The formula essentially computes the arithmetic mean on equidistant points.

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The early form of the Romberg method ?