# Riemann Sums - Function Integration

The purpose of the applet below is to demonstrate how Riemann sums approximate the value of a definite integral. On every subinterval, one can choose either the left or right value of the function, the lower or the larger of the two, or the value at a random point on the interval, or at its midpoint. The applet displays both the definite integral as the function of its upper limit and its approximation by Riemann sums. The function itself can be easily modified by dragging the points on its graph up or down.

As of 2018, Java plugins are not supported by any browsers (find out more). This Wolfram Demonstration, Riemann Sums, shows an item of the same or similar topic, but is different from the original Java applet, named 'Integration'. The originally given instructions may no longer correspond precisely.

Note that in most cases random selection of points on subintervals gives a better approximation than other approaches. Note also how errors of approximation often cancel out to produce a decent final result.

• Cavalieri's Principle
• Derivative of Sine and Cosine
• Distance From a Point to a Straight Line
• Estimating Circumference of a Circle
• Maximum Volume of a Cut Off Box
• Mistrust Intuition of the Infinite
• Naturally Discontinuous Functions
• Rolle's and The Mean Value Theorems
• Function, Derivative and Integral
• Area of a Circle by Rabbi Abraham bar Hiyya Hanasi
• Schwarz Lantern
• Two Circles and a Limit
• Deceptive Appearances
• Problem 4010 from Crux Mathematicorum