The Definite Integral of a Continuous Function

Introduction:

The definite integral of a continuous function f over [a,b] is a limit of sums:

Where:

• P = { x₀, x₁,..., x_n} is a partition of [a,b]
• x _i = x _i - x _i-1
• c _i is in [x _i-1, x_i]
• mesh is the length of the longest section (or sections) in the partition

Approximation by Rectangles:

This applet approximates the value of the definite integral of a continuous function on a given interval by computing the area of a figure made of rectangles. Notice that an area under the x-axis is of a negative value.

To calculate the integral value, I use the Interpreter for symbolic manipulation of mathematical expressions by Jens- Uwe Dolinsky.

Instructions:

1. Enter a continuous function (variable x).
2. Define an interval of the domain.
3. Increase or decrease the number of intervals by pressing >> and/or <<.
4. Choose: midpoint, underestimate, or overestimate.
5. Press "Enter new values" if you choose new function, or new intervals.
6. Compare the approximate values of the Integral and Sum.