Concept

Riemann sum

An approximation to an area built from rectangles, and the thing an integral is the limit of. Its limit as the rectangles narrow is the integral, and whether that limit exists is what integrability means.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

8 rectangles under a curve. A left-endpoint Riemann sum with 8 rectangles approximating the area under a curve.

Adding up rectangles until they stop being rectangles

The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.

analysis · The integral

Area is the undoing of slope

Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.

analysis · The integral

Four staircases against a quarter circle, all of length 2. A quarter circle with staircases of 1, 2, 4, 16 steps drawn over it; each hugs the curve more closely than the last and every one of them is exactly 2 long.

The staircase that is not the diagonal

A staircase can be made to follow a quarter circle as closely as anyone likes. Its length is 2 at every stage and the arc's length is 1.5708, and no amount of refinement closes the gap — which is a fact about length rather than about staircases.

analysis · Arc length

Named alongside it

The objects these essays reach for when they reach for this one.

Limit Area Continuity Convergence Derivative Fundamental theorem Accumulation Antiderivative Approximation Arc length Counterexample Dirichlet

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