Integrals: Definition and How They're Calculated

Picture a curve drawn above a horizontal axis: how much area sits between that curve and the axis, from one point a to another point b? This geometric question is where the definite integral comes from, historically. According to Treccani, the definite integral corresponds exactly to the area of the region between a function’s graph and the x-axis, on an interval [a, b] — one of the central ideas of calculus.

The method behind it is old. Approximating an area with rectangles was already common practice, Fermat among others, long before integral calculus became a formal discipline.

From rectangles to a limit

The practical idea: build n rectangles on the interval, each with its own base, and add up their total area. Wolfram MathWorld calls this sum a Riemann sum — the Riemann integral is the simplest definition of an integral, the one usually met in elementary calculus. The height of each rectangle is the value of the function at a point inside its little interval.

This is where limits come in: as the rectangles get thinner — their width, often written δ, approaching zero — the sum of their areas approaches a single finite value. That limit is exactly what the definite integral is.

Practical example: to find the area under the parabola y = x² between x = 0 and x = 2, OpenStax builds n narrow rectangles under the curve and adds up their areas. As n grows, the rectangles get thinner and the sum gets closer to one precise number. Letting n grow toward infinity and applying the formula for the sum of the first n squares, that number turns out to be exactly 8/3, about 2.667 — the exact area, with no approximation left over.

The fundamental theorem: from antiderivative to derivative

Taking the limit of a sum every single time would be impractical. The fundamental theorem of calculus bridges the definite integral and the indefinite integral — the antiderivative of the function — making it possible to find many areas through an antiderivative, with no limit involved.

For continuous functions, Treccani notes, integrating and differentiating undo each other: finding an antiderivative means retracing, backward, the path that leads from a function to its derivative. The formula that gives the definite integral as F(b) minus F(a), with F standing for one of its antiderivatives, is known as the Newton-Leibniz formula; the same theorem also goes by the name Torricelli-Barrow theorem. On the coordinate plane, this turns into a familiar picture: the area under the curve, evaluated at the two ends of the interval, gives the same answer as the limit of the sums.

The notation that won out

The symbol ∫ still in use today has a precise birthdate. MacTutor reports that on 29 October 1675, Gottfried Wilhelm Leibniz wrote it for the first time, in an unpublished manuscript titled Analyseos tetragonisticae pars secunda, to denote integration. Two weeks later, on 11 November 1675, Leibniz placed the symbol dx next to the integral sign, in the form still familiar today.

Isaac Newton arrived later, by a different route. In his 1704 work Quadratura curvarum, he used a vertical bar above the variable x to mark its integral. Historian Florian Cajori, cited by MacTutor, considered that notation flawed: the bar over the x could be confused with a prime, and the box drawn around the term was difficult to typeset. For that reason, according to Cajori, Newton’s notation never caught on, not even in England — it is Leibniz’s symbol that remained in universal use.