## Quick Reference

Let *f* be a function defined on the closed interval [*a*, *b*]. Take points *x*_{0}, *x*_{1}, *x*_{2},…, *x*_{n} such that *a*=*x*_{0} < *x*_{1} < *x*_{2} <…<*x*_{n−1} < *x*_{n}=*b*, and in each subinterval [*x*_{i}, *x*_{i+1}] take a point *c*_{i}. Form the sumthat is, *f*(*c*_{0})(*x*_{1}−*x*_{0})+*f*(*c*_{1})(*x*_{2}−*x*_{1})+…+*f*(*c*_{n−1})(*x*_{n}−*x*_{n−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.

*Subjects:*
Mathematics.

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