What quantity did you change?
Labs / Mathematics
Riemann Sums
Pick a function, drag the a and b handles to set an interval, and slide the rectangle count up. Each rectangle estimates a sliver of the signed area, and their total creeps toward the exact integral as the strips get thinner. Areas below the axis count as negative.
Labs / Mathematics / Riemann Sums
Riemann Sums
Pick a function, drag the a and b handles to set an interval, and slide the rectangle count up. Each rectangle estimates a sliver of the signed area, and their total creeps toward the exact integral as the strips get thinner. Areas below the axis count as negative.
Why it matters
Why explore this?
Mathematics helps us describe patterns precisely and predict what comes next. This model turns an abstract relationship into something you can change and observe.
Explore
Try the model
approx 0exact 0error 0
Investigate
Questions to test
Predict first, try it in the model, then use what you observe to answer.
What to try
- With only 4 rectangles, which rule—left, right, midpoint, or trapezoid—lands closest to the exact answer?
- Push n from 4 up to 120. How fast does the error fall for the midpoint rule compared with the left rule?
- Slide the interval so it straddles a point where the curve dips below the axis. Why does the total area shrink instead of grow?
- On sin x, drag b out to 2π. Why does the exact integral come out to zero even though there are tall rectangles?
- For a straight-enough piece of a curve, can you make the trapezoid error effectively vanish? What shape of curve fools it the most?
Explain
Make sense of what happened
State the pattern in words, test it with another example, and then connect it to the mathematical rule or representation shown in the model.
Check
Before you leave
What pattern did you observe?
Does the pattern work for another example?
If you can answer all three using evidence from the model, you understand the main idea. If not, return to Explore and change one setting at a time.