Rectangles that fill the area under a curve, area as a function that grows, and the theorem that says slope and area undo each other
A derivative answers how fast something is changing. Integration turns the question around: given how fast something changes, how much has it changed in total? The answer is an area, found by adding up thin rectangles, and the fundamental theorem of calculus reveals that finding areas and finding slopes are inverse operations.
Choose a function and drag a and b. Increase the number of rectangles, switch between sampling rules, and press refine to keep doubling n. The small plot on the right shows every rule closing in on the same number.
Drag a and b along the axis (b may pass a: then the integral changes sign). Upper and lower sums take the largest and smallest of 17 samples in each strip, which is exact for monotone pieces and very close otherwise.
Split [a, b] into n strips of width Δx = (b − a)/n, pick a sample point xᵢ* in each, and add up the signed areas of the rectangles:
Sₙ = ∑ᵢ f(xᵢ*)·Δx → ∫ₐᵇ f(x) dx as n → ∞.
For a continuous f the limit exists and does not depend on how the sample points are chosen; that common value is the definite integral. Area below the axis counts as negative, so ∫ₐᵇ f is a net signed area. For x² on [0, 1] the right sum is (n + 1)(2n + 1)/(6n²), which tends to 1/3.
The rules differ in how fast they converge. Left and right sums have errors proportional to 1/n; the midpoint and trapezoid rules, which balance the overshoot on one side of each strip against the undershoot on the other, have errors proportional to 1/n². Upper and lower sums, using the largest and smallest value in each strip, trap the integral between them for every n.
Archimedes found the area under a parabola by exhaustion, filling it with ever smaller triangles, around 250 BC. Bernhard Riemann gave the modern definition of the integral as a limit of such sums in 1854, and Gaston Darboux reformulated it with upper and lower sums in 1875.
Let the right-hand end of the area move. Press sweep and watch A(x), the area from a to x, being drawn underneath: rising where f is positive, falling where it is negative, with a tangent whose slope is always f(x).
Drag in either panel to move x; drag the amber diamond to move the base point a. Moving a changes A by a constant (the area between the old and new base points), so the shape of A, and its slope, never change.
Fix a base point a and let the upper limit vary. The result is a new function of x:
A(x) = ∫ₐˣ f(t) dt.
The variable of integration is renamed t so that it cannot be confused with the endpoint x. A(a) = 0, A increases on intervals where f > 0 and decreases where f < 0, and A has a local maximum wherever f changes sign from positive to negative. Its concavity follows f′: where f is increasing, A is concave up.
Some important functions are defined this way. With f(t) = 1/t and a = 1, A(x) is the natural logarithm ln x. With f(t) = e^(−t²) and a = 0, A(x) is (up to the factor 2/√π) the error function of statistics, which has no formula in terms of elementary functions; the accumulation function is its definition.
Moving the base point from a to a′ changes A by the constant ∫ₐ′ᵃ f(t) dt, so all these functions have the same shape and the same slope. That constant is the “+ C” attached to every antiderivative.
Part 1: shrink the strip width h and watch the new sliver of area become a rectangle of height f(x). Part 2: read f as a velocity, add up the rectangles as distances, and see the running total land exactly on F(b) − F(a).
A(x) = ∫ₐˣ f(t) dt with a = 0. The strip’s area is computed by adaptive Simpson integration; the magnified panel always shows the strip at the same width on screen.
Part 1. If f is continuous, then A(x) = ∫ₐˣ f(t) dt is differentiable and A′(x) = f(x). The proof is the picture: A(x + h) − A(x) is the area of a strip of width h, trapped between h times the smallest and h times the largest value of f on [x, x + h]. Dividing by h and letting h → 0, continuity squeezes the quotient to f(x).
Part 2. If F is any antiderivative of f (F′ = f), then
∫ₐᵇ f(x) dx = F(b) − F(a).
Proof: A and F have the same derivative, so by the mean value theorem they differ by a constant, F = A + C. Then F(b) − F(a) = A(b) − A(a) = A(b). In the language of motion: if f is velocity, each rectangle f(tᵢ)Δt is a short distance, their sum is the total displacement, and the position F records exactly that, so the trip meter at the end minus the reading at the start equals the area under the velocity curve.
A limit of sums, which would be hard to compute, becomes two evaluations of an antiderivative: ∫₀² x² dx = 2³/3 − 0 = 8/3, and ∫₀^π sin x dx = −cos π + cos 0 = 2. Isaac Barrow published a geometric version in 1670; his Cambridge successor Isaac Newton and, independently, Gottfried Wilhelm Leibniz made it the engine of calculus, turning areas, volumes and distances into problems about antiderivatives.
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