Riemann Integral

Objective

The objective is to understand the rationale behind Riemann integrals by approximating area under a curve.

Base Difficulty

Hard in concept, medium in execution

Concept

The student should research on Riemann sums, which is the simplest way to approximate an integral. Click here for the Riemann Integral Wikipedia page.

Method of Remann Integration

In the picture above, red corresponds to left endpoint positioning, white corresponds to right endpoint positioning, and the midpoint positioning should be the average of red and white.

Materials

Procedure

Analysis

Compare the estimated area from experiment against the exact area of the shapes. The exact area could be obtained in two ways: advanced students with pre-calculus knowledge could go ahead and evaluate the definite integral from the corresponding function, by hand or using a calculator/computer. An alternative is to carefully weigh a thick cardboard of known dimensions, and then finely cut the shape out of the cardboard, weigh the cut shape and calculate its area from weight ratios. Comment on the difference between left, right, and midpoint positioning. Comment on the role played by strip width.

Conclusions

The conclusion drawn will depend on the hypothesis chosen, but a comparison between the approximated values and the actual values of the area should play a part.

Extensions

The same method of approximation could be used to try to calculate the value of pi (with a circle of known radius).