The generalized method of exhaustion (Q1607954)

The author uses trapezia defined by equi-interval partitions to approximate the area under a piece-wise continuous curve. By dividing \([a,b]\) into \(2, 4, \dots, 2^m, \dots\) equal intervals and using these approximations the author obtains the formula \[ \displaystyle{{1}\over{b-a }}\int_a^b f = \sum_{n=1}^{\infty}\sum_{m=1}^{2^n-1} {{(-1)^{m+1}}\over{2^n}}f\bigl(a+ m(b-a)/2^n\bigr). \] Applications are made to give series expansions of various functions; for instance \[ \displaystyle{ {\sin x}\over {x} }= \sum_{n=1}^{\infty}\sum_{m=1}^{2^n-1} {{(-1)^{m+1}}\over{2^m}}\cos (mx/2^n) \] {}.