Build accumulation from small contributions

Try

A pipe delivers water at a changing rate. During a short interval, multiplying a representative flow rate by the interval duration estimates the water delivered. If the rate is 4 litres per minute and the duration is 0.2 minutes, that contribution is 0.8 litres. Explain why adding rates alone would have the wrong units for volume.

Predict

For f(x)=x² on [0,2], predict whether rectangles using left endpoints lie below or above the curve. Make the prediction from monotonicity, not from a memorized rule: on this interval the function increases. On an interval where the function decreases, the answer reverses.

Experiment

Live experiment: from slices to an integral

Predict whether left rectangles underestimate x² on this interval. Compare two slices with four, then switch the rule. Record the signed error, not just whether two rounded estimates look alike.

Function x² on [0, 2]x³ − x on [−1, 1] Rule Left rectanglesMidpoint rectanglesTrapezoids Number of equal slices 24681224 Reset experiment

Width: Δx = (2 − (0)) / 2 = 1.

Estimate: 1. Exact signed integral: 8/3. Estimate minus exact: -1.666667.

Inspect every contribution

Interval	Sample height	Signed contribution
[0, 1]	0	0
[1, 2]	1	1

Displayed decimal estimates and contributions are rounded to six places. These are known smooth teaching functions. A finite estimate is not a general proof of convergence. The blue curve uses 160 drawing segments; panel contributions use the model values. Settings are temporary and do not add completion credit.

Use the live controls to compare sample choices at a fixed slice count, then refine one rule. The live investigation labels its own interval and values; the following table is a separate worked example on [0,2].

Divide [0,2] into two intervals of width 1. Compare three choices of sample point, then draw their rectangles on the same coordinate scale.

Rule	Sample inputs	Heights	Sum
Left	0, 1	0, 1	1
Right	1, 2	1, 4	5
Midpoint	1/2, 3/2	1/4, 9/4	5/2

Repeat with four equal intervals. Left and right sums become 7/4 and 15/4. The midpoint sum becomes 21/8. A later theorem will give the exact integral 8/3; compare these fractions without rounding them prematurely. The quadrature investigation lets you compare the bounded numerical rules, but a plot or one refinement is not a proof of convergence.

Understand

A partition divides [a,b] into subintervals with endpoints x₀=a through xₙ=b. Their widths are Δxᵢ=xᵢ−xᵢ₋₁. Choose a sample point ξᵢ in each subinterval. The Riemann sum is the sum of f(ξᵢ)Δxᵢ. Widths need not be equal, and sample points need not be endpoints. Every product represents one signed contribution.

The definite Riemann integral is the common limiting value of these sums as the largest subinterval width tends to zero, when that limit exists independently of the choices. This largest width is the mesh. Merely adding more intervals is insufficient if one large interval remains unrefined. Continuous functions on closed bounded intervals are Riemann integrable; a full proof requires control of variation on small intervals, developed in analysis.

For equal right rectangles under x² on [0,2], width is 2/n and sample input is 2i/n. Thus the sum is (8/n³) times the sum of i² from i=1 to n. Using the established identity Σi²=n(n+1)(2n+1)/6 gives (4/3)(1+1/n)(2+1/n), which approaches 8/3. This establishes a particular sequence’s limit. Continuity supplies the broader integrability theorem ensuring other valid refining choices agree.

Finite measurements need separate interpretation. Four observed rates do not reveal all behavior between observations. Linear interpolation gives a definite mathematical curve and a trapezoid estimate for that curve, but another continuous rate could share the observations and have a different total. Distinguish numerical approximation error for a specified function from uncertainty about the function itself.

Apply

Write sample points, widths and units before calculating each practice sum. For the proof task, retain n until after the algebraic simplification. For bounds, name the monotonicity assumption. For measured data, state what interpolation assumption turns readings into a model.

Continue

Next organize signed contributions using the properties of definite integrals. A negative contribution can represent net outflow or motion in the negative direction; it does not make the total geometric area negative.