Riemann Sums - Negative Integrals and Discontinuities

This is an extension of the Riemann Sums applet we met earlier.

In the applet below, you can explore the concept of numerical integration when negative integrals and discontinuities are involved.

Below you'll see some of the integrals we solve in this chapter.

Things to Do

In this applet, you start with a predefined function that has been drawn for you. You can:

• Use the bottom slider below the curve to change the number of intervals.
• Now try different options from the "Choose Riemann Sum type" pull-down menu.
• You can change the start and end `x`-values using the slider with the green bar.
• You can choose different example functions from the pull-down menu.
• Observe how "negative areas" (rectangles or other shapes below the `x`-axis) affect the integral value.
• In cases with discontinuities, observe that we can integrate on one side or the other of the discontinuity, but not across it. (The rectangles still appear near the discontinuities.)

The ideas of area under a curve and integration are closely linked. Be careful if you need to find actual areas, that you take the absolute value of the integral for portions under the `x`-axis!

[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.]