6a. Riemann Sums Applet
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:
- Use the blue 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 and why.
- You can change the start and end x-values.
- 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.
Choose Riemann Sum type:
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Sum of areas
For comparison, the actual integral value:
We can see the parabolas that approximate the curve across each pair of segments. The parabolas appear at 1 second intervals. The number of divisions has to be even for Simpson's Rule.
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 this concept. 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.