Let f be a function defined on the closed interval [a, b]. Take points x0, x1, x2,…, xn such that a=x0 < x1 < x2 <…<xn−1 < xn=b, and in each subinterval [xi, xi+1] take a point ci. Form the sumthat is, f(c0)(x1−x0)+f(c1)(x2−x1)+…+f(cn−1)(xn−xn−1). Such a sum is called a Riemann sum for f over [a, b]. Geometrically, it gives the sum of the areas of n rectangles, and is an approximation to the area under the curve y=f(x) between x=a and x=b.
The (Riemann) integral of f over [a, b] is defined to be the limit I (in a sense that needs more clarification than can be given here) of such a Riemann sum as n, the number of points, increases and the size of the subintervals gets smaller. The value of I is denoted bywhere it is immaterial what letter, such as x or t, is used in the integral. The intention is that the value of the integral is equal to what is intuitively understood to be the area under the curve y=f(x). Such a limit does not always exist, but it can be proved that it does if, for example, f is a continuous function on [a, b].
If f is continuous on [a, b] and F is defined bythen F′(x)=f(x) for all x in [a, b], so that F is an antiderivative of f. Moreover, if an antiderivative ϕ of f is known, the integralcan be easily evaluated: the Fundamental Theorem of Calculus gives its value as ϕ(b)−ϕ(a). Of the two integralsthe first, with limits, is called a definite integral and the second, which denotes an antiderivative of f, is an indefinite integral.