Visualizing the Techniques for Numerical Integration

Play with the applet to get a feel for it.

• It is color coordinated; things with the same color are related.
• Can you see how the yellow, red or orange objects are related to each other?
• Can you see how the yellow, red or orange ojbects are related to the cyan graph?
• How is the graph on the right affected by the different tools in the applet?

Follow the instructions below to explore Riemann sums and other tecnhiques of numerical integration.

Number of subdivisions in Riemann sums

1. Use the pull-down menu to select the function f(x) = x² - 2x. Use the sliders on the right to set n equal to 1 and the evaluation point to the left endpoint.
2. Record the height of the yellow dot displayed on the right side of the screen. This is an estimate of the value of the integral of f(x) = x² - 2x from -1 to 2.
3. The estimate equals the area of the yellow rectangle shown on the left. Do you think that this is a good approximation of the (signed) area between the blue curve and the x axis?
4. Adjust the slider under the right graph so that n = 3. Record the estimate shown on the right graph.
5. Judging by the rectangles shown on the left, do you think this is a more or less accurate estimate than the one for n = 1?
6. Obtain estimates for n = 6, n ≅ 12 and n = 500. Which do you think is most accurate?
7. If you can, calculate the exact value of the integral and compare it to the numerical estimates you obtained.

Accuracy of Riemann sums

1. Use the slider on the right to set n equal to 6.
2. Use the figure on the left to compare the four options Min, Max, Evaluation point: Left endpoint and Evaluation point: Right endpoint.
3. Can you describe the rule used to compute the Min and Max estimates?
4. Use the slider on the right to set n equal to 500. A dotted curve appears in the right hand figure.
5. Compare the graph in the right hand figure for the four options Min, Max, Evaluation point: Left endpoint and Evaluation point: Right endpoint.
6. If you like, observe how the graph in the right hand figure changes when you adjust the Evaluation point slider.
7. Assuming that as the value of Δx decreases the accuracy of the numerical estimate increases, the height of the left end of the dotted curve should be close to the exact value of the integral. Based on this assumption, which of the four methods seems to be the most accurate?
8. If you like, use the pull-down menu on the left to repeat this experiment for other functions f(x).
9. The Max and Min estimates seem less accurate than the Evaluation point estimates. Can you think of a situation in which it would be more useful to have a Max or Min estimate?

Other Techniques

1. For each of the function options in the pull-down menu on the left:
   • Set n = 2.
   • Study the figure on the left with Evaluation Point (Riemann sum) selected. This illustrates the technique used to obtain a Riemann sum approximation.
   • Study the figure on the left with Trapezoidal selected to see the regions whose areas are summed when using the trapezoidal rule.
   • Check the box next to "Simpson's" to see the regions whose areas are used when applying Simpson's rule.
   • Which of these three techniques comes closest to computing the exact area under the curve?
2. Set n = 500.
3. For the functions provided in this mathlet, the the dotted curve on the right will be closer to a horizontal line for more accurate techniques of approximation. (Why?)
4. For each of the functions available, compare the three techniques. Is one technique always most accurate? Is one technique always least accurate?