This will be useful when we cannot find an elementary antiderivative for f or if the function is defined using data obtained from some experiment. Let's use the notation introduced in the summary on Riemann Sums where we discussed three specific approximations of a definite integral:

1. Riemann sums using left-hand endpoints
   

2. Riemann sums using right-hand endpoints
   

3. Riemann sums using midpoints.
   

Trapezoidal Rule:

• Discussion [Using Flash] [Using Java]

• An animation showing graphically how the Trapezidal Rule is an approximation of the definite integral.

• Example #1.

• Using the TI-85 graphing calculator to approximate a definite integral using the Trapezoidal Rule.

• The Trapezoidal Rule and the Midpoint Rule - under-estimates and over-estimates.
  Discussion [Using Flash] [Using Java]

• Using Maple to approximate a definite integral using the Trapezoidal Rule.

Simpson's Rule:

• Discussion [Using Flash] [Using Java]

• Example #2.

• Simpson's Rule and approximations by parabolas.
  Discussion [Using Flash] [Using Java]

• An animation showing graphically how Simpson's Rule is an approximation of the definite integral.

• Using the TI-85 graphing calculator to approximate a definite integral using Simpson's Rule.

• Using Maple to approximate a definite integral using Simpson's Rule.

Error Bounds.

TI-85 graphing calculator programs for numerical approximation of integrals.