The numerical procedures developed in this chapter for evaluating integrals have all used strips of equal width. An alternative procedure is to specify the number of tabular points to be used but not their position. It is possible to find tabular points within the domain of integration for the most accurate evaluation of the integral for the given number of points. Consider the two-point formula

where a, b, α and β are constants to be found. Symmetry about x = 0 implies β = –α. If the formula evaluates all quadratic functions exactly, prove that

'h -h f(x)dx = h[af(ah) + bf (Bh)]