Integrals turn continuous summing into area and volume
An integral is the continuous counterpart of a sum, used to find areas, volumes, and their generalizations, and integration is one of calculus’s two basic operations alongside differentiation. Newton and Leibniz formulated its principles independently in the late 17th century, picturing area as infinitely many thin rectangles.
Democritus and Eudoxus in the 4th century BC used the method of exhaustion, carving a shape into infinitely many pieces of known size. Archimedes pushed it further in the 3rd century BC to find the area of a circle, the surface and volume of a sphere, the area under a parabola, and the area of a spiral. Liu Hui developed a similar approach in China around the 3rd century AD, and in the 5th century Zu Chongzhi and his son Zu Geng used it for the volume of a sphere. Ibn al-Haytham later devised sums of squares and fourth powers to find the volume of a rotated parabola.
In the 17th century Cavalieri’s method of indivisibles handled powers of x up to the ninth, and the case of 1/x needed a new function, the hyperbolic logarithm, found in 1647. Barrow and Torricelli glimpsed the link with differentiation, Barrow gave the first proof of the fundamental theorem, and Wallis extended the method to negative and fractional powers. Leibniz introduced the ∫ sign in 1675, an elongated s standing for summa, while Newton’s bar and box notations proved easy to confuse and hard to print. Jacob Bernoulli first printed the term in Latin in 1690, and Fourier began writing limits above and below the sign around 1819–1820.
Bishop Berkeley mocked Newton’s vanishing increments as ghosts of departed quantities. Rigour arrived with limits: Bernhard Riemann defined the integral using ever thinner vertical slabs, and in the early 20th century Henri Lebesgue built a more general version on measure theory, prompted partly by functions from Fourier analysis that Riemann’s definition could not handle.
A worked example shows the idea. Estimating the area under √x from 0 to 1 with five rectangles using right-hand heights gives about 0.7497, too high, while twelve left-hand rectangles give 0.6203, too low; as the pieces multiply, the sums close in on exactly 2/3. Line integrals replace the interval with a curve, and surface integrals replace it with a patch of surface.
Source: Integral
