Archimedes’ method
Archimedes of Syracus lived 287 to 212 B.C. and was one of the most important mathematicians of his time. He found a method to calculate the exact area under the curve of a parabola. It was not until 1630 when Cavalieri continued work in this area of mathematics, followed by Newton and Leibniz around 1670 who developed calculus. The “strip method”, Archimedes uses, is fundamental for understanding the ideas involved in integration.
Find an estimate of the area between the graph of the function $f(x) = x^2$
and the x-axis.
Here, the interval is divided into 4 equal strips. In the first diagram, the height is always equal to the lowest value of f(x) in that interval while in the second diagram the height is equal to the maximum value of f(x) in that interval. These are the lower sum (inferior Riemann sum) and the upper sum (superior Riemann sum), respectively (Ger. Untersumme und Obersumme).
Task 1
Calculate the lower sum area and the upper sum area. Find the average of these sums.
solution video
Task 2
Now apply the same method to 8 intervals.
Mid-Ordinate Rule
This is a similar approach to that of Archimedes. Here a series of rectangles, usually of equal width, is used to determine an approximate area beneath a curve. The height of the rectangle is the value of f(x0) where x0 is the mid-value of each rectangle.
Trapezium Rule
Simpson’s Rule
TODO
Kepler’s Rule
TODO
links
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geogebra applets by Matthias Hornof, Germany ↩︎