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.
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
- Construction paper
- Ruler
- Straightedge (possibly another ruler)
- Scissor
- Print-outs of images of different curves from our CPS Science Fair Bank website (click here to download)
Procedure
- Cut the construction paper into rectangular strips - the same width for simplicity, or differing widths for a more complicated hypothesis.
- Lay the strips of construction paper across the curve (give an example image) following the left, right, or midpoint positioning.
- Placing the straightedge across the bottom of the curve and above the construction paper strips may help with measurement.
- Sum the areas of the rectangles to find the approximate area under the curve!
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).