The integral is a mathematical operation that finds the area under a curve, representing accumulation of a quantity.
Integration is the reverse process of differentiation. The definite integral β«βα΅ f(x) dx computes the net area between f(x) and the x-axis from x=a to x=b. Integration is used to compute areas, volumes, work, center of mass, and accumulated change. The Fundamental Theorem of Calculus connects derivatives and integrals, stating that differentiation and integration are inverse operations. Techniques include substitution, integration by parts, partial fractions, and trigonometric substitution.
Integration originated in ancient times with Archimedes' method of exhaustion for finding areas. Modern integral calculus was developed by Isaac Newton and Gottfried Leibniz in the 17th century. The integral symbol β« was introduced by Leibniz in 1675, representing a long "S" for "summa" (sum). Bernhard Riemann formalized the Riemann integral in the 19th century, and later Henri Lebesgue developed the more general Lebesgue integral.