Suppose f (x) is a function of x on a finite interval [a, b]. Recall that the definite integral of f on [a, b] can be defined by the limit of Riemann sums, if the limit exists: If a= x0 < x1 < . . . < xn = b is a partition of [a, b], x∗k are sampling points satisfying xk−1 ≤ x∗k ≤ xk for k= 1, 2, . . . ,n, and xk = xk − xk−1 for k= 1, 2, . . . ,n, then

k=1 f (x∗k) xk (7.1)

is a Riemann sum. The xk’s don’t have to be equal. If the limit of the Riemann sums exists∗ as n → ∞ and max

1≤k≤n | xk | → 0, we define the definite integral of f from x= a

to x= b: b a

f (x)dx = lim n→∞

( n∑


. (7.2)

If the definite integral of f on [a, b] exists and f (x) ≥ 0 there, then ba f (x)dx equals the area under the curve y= f (x) and above the interval a ≤ x ≤ b on the x-axis, as illustrated in Figure 7.1.