Riemann Sums and the Integral Simulation

The area under a curve can be approximated by slicing it into strips and adding up their areas. Each strip has to pick a height — its left edge, its right edge, its middle, or a sloping top joining both — and each choice gives a different answer. As the strips get narrower every choice converges on the same number, and that limit is the definite integral. Plotting the running total as its own curve shows the area function, whose slope is the function you started with. Students can change number of strips and from and compare what happens. Use it for math inquiry with High School on a projector or student device.

Standards: Common Core CCSS.MATH.CONTENT.HSA.REI.D.11, CCSS.MATH.CONTENT.HSF.IF.B.6

• Compare the left and right rules on a rising curve and see one undershoot while the other overshoots
• Double the number of strips and watch what the error does each time
• Find out why the midpoint rule beats the trapezium rule, even though its tops are flat
• Move a and b and watch the exact area change with them
• Look at the running-total curve and see that its steepness follows the height of the original function

1. 1Predict how changing number of strips will affect the strip total, then test the prediction.
2. 2Use evidence from the live calculation and visual to explain the result.

Questions teachers ask