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Concept

Integral

It is the limit of sums of rectangles under a curve, and it measures accumulated total rather than instantaneous rate. The fundamental theorem says that it undoes differentiation, which is what makes areas computable from antiderivatives.

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

50001000015000200001.001.051.101.151.20xπ(x) ÷ the estimate÷ x / ln x÷ li(x)an exact estimate would sit herethe ratio of π(x) to each of its two estimates, from 100 to 20,000; a ratio of one means the estimate is exactπ(100) = 25 · x / ln x says 22 · li(x) says 29 π(1,000) = 168 · x / ln x says 145 · li(x) says 177 π(10,000) =1229 · x / ln x says 1086 · li(x) says 1246

Counting what has no formula

There is no expression that gives the nth prime, and yet the number of primes below a bound is predictable to within a fraction of a per cent — by a function that is not a formula for the primes but an integral of the wrong-looking quantity.

number · prime distribution uv = 1.66v·Δuu·ΔvΔu·Δv = 0.300u = 1.40Δuv = 1.18Δvhcorner's sharequotient0.900016.38%2.03510.45009.26%1.91060.22504.96%1.84430.11252.58%1.81000.05631.31%1.79250.02810.66%1.7837u'v + uv' = 1.7748u = x and v = √x at x = 1.4, grown by h = 0.9: the increment 1.8316 is 1.0649 + 0.4667 + 0.3000, and the last of thoseis the cornerhalving h six times takes the corner from 16.4% of the increment to 0.66%, while the quotient goes 2.035 → 1.784against u'v + uv' = 1.775

A rectangle grown on two sides

A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.

analysis · the derivative

Named alongside it

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

ApproximationAreaConvergence rateCounting argumentDensityDerivativeDifference quotientDissectionLimitLinearityLogarithmNatural logarithm

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