The principles of integration were formulated independently by Isaac Newton and Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width. This definition does not give additivity. [[Darboux Integrable Function|Riemann]] later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalised Riemann's formulation by introducing what is now referred to as the [[Lebesgue Integral]]; it is more general than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalised depending on the type of the function as well as the domain over which the integration is performed. For example, a [[Line Integral of Vector Field on Subset of Real n-Space|line integral]] is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting two points in space. In a [[Integration in Cartesian coordinates|surface integral]], the curve is replaced by a piece of a surface in 3d. # References 1. Wikipedia, Integral. https://en.wikipedia.org/wiki/Integral.