Let f be a continuous function whose domain includes the closed interval [a, b]. We have investigated ways of approximating the definite integral

We are now interested in determining how good are these approximations. We define the error:

  1. Riemann sums using left-hand endpoints:

  2. Riemann sums using right-hand endpoints:

  3. Riemann sums using midpoints:

  4. Trapezoidal Rule:

  5. Simpson's Rule:

Trapezoidal Rule Error Bound: Suppose that the second derivative f'' is continuous on [a, b] and suppose that |f''(x)| < M for all x in [a, b]. Then

Simpson's Rule Error Bound: Suppose that the fourth derivative f'' is continuous on [a, b] and suppose that |f(4)(x)| < M for all x in [a, b]. Then

Midpoint Rule Error Bound: Suppose that the second derivative f'' is continuous on [a, b] and suppose that |f''(x)| < M for all x in [a, b]. Then