How Do Rectangles Reveal the Area Under a Curve?

Introduce the driving question: how can we measure the area under a curve when it doesn't have straight edges? Pose the tension that rectangles only approximate, leaving open whether the limit gives an exact answer.

• Curved regions can't be measured with a ruler or simple shape formulas.
• Rectangles give a rough estimate — but is the estimate ever exact?
• Sets up the central question the investigation will answer.

Learner commits to an intuition about whether Riemann sums can be made exact and whether the choice of rectangle height matters.

• Predictions are committed before evidence is shown.
• Tests intuition about convergence and sampling independence.

Manipulable simulation where the learner adjusts the number of rectangles under a curve and toggles between left-endpoint, right-endpoint, and midpoint sampling to see the sum change.

• Slider changes the number of rectangles from coarse to fine.
• Toggle switches the sampling rule: left, right, or midpoint.
• Numerical sum and a visual shaded region update live.
• Different sampling rules give different sums at low N, but converge as N grows.

Explain that as the partition width shrinks, the difference between any two reasonable Riemann sums vanishes. The definite integral is defined as this common limit, independent of sampling choice.

• Partition width Δx controls rectangle thinness.
• For a continuous curve, the maximum height variation inside each strip shrinks as Δx → 0.
• The limit of the sum is the definite integral — a single well-defined value.
• Sampling rule affects intermediate approximations, not the limit.

Interactive comparison showing what happens when the integrand has a jump discontinuity: two sampling rules no longer converge to the same value, revealing when Riemann sums are well-defined.

• Switch from a continuous curve to a step function with a jump.
• Left and right endpoint sums converge to different values.
• Illustrates that Riemann integrability requires sufficient continuity.

Learner tests a different function in the same simulator, predicting and then observing whether the same convergence behavior holds for a curve with varying steepness.

• Switch between curves to test the principle on a new shape.
• Reuse the partition slider to confirm convergence for the new curve.
• Generalizes the conclusion beyond the original example.

Directly answer the driving question: the Riemann sum converges to a single value as rectangles become infinitesimally thin, and that value is the definite integral — the exact area under the curve.

• Recap: thin rectangles approximate, the limit gives exactness.
• Sampling rule does not change the limit for continuous functions.
• Connects the visual rectangles to the symbol ∫ f(x) dx.
• Sets up the next step: how to evaluate this limit efficiently.