Concept

Antiderivative

A function whose derivative is a given function, determined only up to an added constant. By the fundamental theorem of calculus, an antiderivative evaluated at the two ends of an interval gives the area under the curve between them.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

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

Integration by parts is a rectangle. The increasing curve v = u²/4 between u = 1 and u = 3. The region under it is shaded one way and the region between it and the vertical axis another; together they fill the rectangle from the origin to (3, 2.25) minus the rectangle to (1, 0.25).

A rectangle cut by a curve

Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.

analysis · The integral

Named alongside it

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

Area Fundamental theorem Accumulation Continuity Derivative Factorial Inequality Integral Inverse function Limit Logarithm Mean value theorem

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