The second important kind of limit encountered in the calculus is the limit of a sum of elements when the number of such elements increases without bound while the size of the elements diminishes. For example, consider the problem of determining the area under a given curve *y* = *f* ( *x* ) between two values of *x,* say *a* and *b.* Let the interval between *a* and *b* be divided into *n* subintervals, from *a* = *x* _{0} through *x* 1, *x* 2, *x* 3, … *x* i - 1,

This approximation can be improved by increasing *n,* the number of subintervals, thus decreasing the widths of the Δ *x* 's and the amounts by which the Δ *A* 's exceed or fall short of the actual area under the curve. In the limit where *n* approaches infinity (and the largest Δ *x* approaches zero), the sum is equal to the area under the curve:;e4;none;1;e4;;;block;;;;no;1;71808n;113795n;;;;;eq4;comptd;;center;stack;;;;;CE5The last expression on the right is called the integral of *f* ( *x* ), and *f* ( *x* ) itself is called the integrand. This method of finding the limit of a sum can be used to determine the lengths of curves, the areas bounded by curves, and the volumes of solids bounded by curved surfaces, and to solve other similar problems.

An entirely different consideration of the problem of finding the area under a curve leads to a means of evaluating the integral. It can be shown that if *F* ( *x* ) is a function whose derivative is *f* ( *x* ), then the area under the graph of *y* = *f* ( *x* ) between *a* and *b* is equal to *F* ( *b* ) - *F* ( *a* ). This connection between the integral and the derivative is known as the Fundamental Theorem of the Calculus. Stated in symbols:;e5;none;1;e5;;;block;;;;no;1;71808n;168913n;;;;;eq5;comptd;;center;stack;;;;;CE5The function *F* ( *x* ), which is equal to the integral of *f* ( *x* ), is sometimes called an antiderivative of *f* ( *x* ), while the process of finding *F* ( *x* ) from *f* ( *x* ) is called integration or antidifferentiation. The branch of calculus concerned with both the integral as the limit of a sum and the integral as the antiderivative of a function is known as the integral calculus. The type of integral just discussed, in which the limits of integration, *a* and *b,* are specified, is called a definite integral. If no limits are specified, the expression is an indefinite integral. In such a case, the function *F* ( *x* ) resulting from integration is determined only to within the addition of an arbitrary constant *C,* since in computing the derivative any constant terms having derivatives equal to zero are lost; the expression for the indefinite integral of *f* ( *x* ) is*C* must be determined from various boundary conditions surrounding the particular problem in which the integral occurs. The calculus has been developed to treat not only functions of a single variable, e.g., *x* or *t,* but also functions of several variables. For example, if *z* = *f* ( *x,y* ) is a function of two independent variables, *x* and *y,* then two different derivatives can be determined, one with respect to each of the independent variables. These are denoted by ∂ *z* /∂ *x* and ∂ *z* /∂ *y* or by *D* _{ x } *z* and *D* y

- Introduction
- The Differential Calculus
- The Integral Calculus
- Bibliography

*The Columbia Electronic Encyclopedia,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.