6a. Riemann Sums Applet

You can use this applet to explore the concept of numerical integration. We met this concept before in Trapezoidal Rule and Simpson's Rule.

Before integration was developed, the only way to find the area under a curve was to draw rectangles with increasingly smaller widths to get a good approximation.

Remember, we are using the area under a graph to represent some physical quantity. For example, integration can help us to find a velocity from an acceleration, or to solve problems in electronics.

Things to Do

In this applet, you start with a predefined function that has been drawn for you. You can see calculation changes in the Information section as you:

• Use the first slider below the curve to change the domain of the graph. Observe the changes in the Information section underneath the graph sliders. to the right of the graph.
• Use the second slider below the curve to increase the number of intervals (try n = 20).
• Now try different options from the "Choose Riemann Sum type" pull-down menu. Consider which one gives the best approximation for each example and why.
• You can choose different example functions from the pull-down menu at the top.

Note: You have not seen how to integrate some of these examples yet. We'll learn how to do them in a later chapter. It was like this in Isaac Newton's day - they could find approximations for their integrals using this method, but were desperately trying to find a better algebraic way to do it. Computers, of course, use numerical methods to find definite integrals.

Integration is an extension of this idea. If we take infinitely many rectangles, and make them infinitely thin, we get the "exact" area under the curve.

Recall that there is a wide range of practical problems that are solved using integration. It's not just for finding areas under curves!

More advanced applet

The above applet has continuous function examples, where the curve is completely above the x-axis for all values of x. You can see a more advanced applet that has curves with negative values and discontinuities here: Riemann Sums applet - negatives and discontinuities.

[Credits: Based on a combination of JSXGraph Riemann sum III and a Java applet by David Eck and team from the Hobart and William Smith Colleges.]