The definite integral
a
b z f(x) dx is defined as the limits of the sum f(x1)δx1 + f(x2)δx2 + ⋅⋅⋅ +
f(xn)δxn when n → ∞ and each of the lengths δx1, δx2, ..., δxn tends to zero. Here δx1, δx2, ..., δxn
are n sub-intervals into which the range b – a has been divided and x1, x2, ..., xn are values of
x lying respectively in the first, second, ..., nth sub-interval.