Integrals
Integrals are the area of mathematics that deals with the integration of a function. The area of mathematics is concerned with the subject of the integrals themselves. Integrals are usually written on the right side of an equal sign. Integrals are also used in a variety of formulas, particularly in mathematics and physics. Integrals are usually written using a definite integral symbol (for example, ) or a function notation (for example, sin(x)dx, which is the integral of the sine function with respect to the variable x). The definite integral of a function f(x) with respect to a variable x is denoted by: or Integrals can be classified into two areas: The area called elementary calculus is devoted to the integration of elementary functions such as polynomials, rational functions, trigonometric and exponential functions. The area called advanced calculus is devoted to the integration of more complicated functions such as exponential and logarithmic functions, trigonometric and hyperbolic functions, and even functions of several variables. The remainder of this section discusses elementary functions. The indefinite integral and the definite integral are two different forms of integration. The definite integral is the standard form of integration used in calculus. The definite integral may or may not be unique; if two functions have the same definite integral, then they are equal. The indefinite integral is the integral of a function in which the limits of integration are not fixed. For example, the indefinite integral of (ex) is In the above integral, the limit of integration is infinity, but it is not specified. The indefinite integral of a function is unique if the function is continuous and its limits of integration are in the closed interval [a, b] (a closed interval is an interval with a definite end point, such as (2, 3), or [0, ?)), or if the function is differentiable with a derivative at every point of the closed interval. If it is not differentiable at every point in a closed interval, the indefinite integral of the function with respect to any variable is undefined. The indefinite integral of a function is a special case of the integral of the integral. Using the Riemann sum theorem, the indefinite integral of a function "f"("x") is given by: The fundamental theorem of calculus states that the indefinite integral of a function of a real variable over a closed interval containing the integral of the function is equal to the integral of the function of the function. The result of this theorem can be extended to the case of an unbounded interval containing the integral of the function. Where the function is defined at an open interval containing the integral, then the result of the extension is the same as the result of the theorem. The definite integral is an integral over a closed interval of the form: where "a" and "b" are real numbers and the function "f" is continuous on the closed interval. The left-hand side of the definite integral can be interpreted as the area between the curve "y" = "f"("x") and the x-axis, and the right-hand side as the area between the curve "y" = "f"("x") and the x-axis. For the function formula_3, the definite integral can be calculated as follows: The integral of a function of a single variable is the limit of a sequence of approximations of the integral of the function. The limit of a finite sequence of approximations is the same as the integral of the function. The following table shows an infinite sequence of approximations of the integral of sin ("x"): Each of these terms is a quotient of the previous term by the difference between the limits of the interval. The limit of a finite sequence of approximations, if finite, is the same as the integral of the function. The limit of an infinite sequence of approximations is a function of the limits of the sequence. The limit of the sequence is therefore the original function. As an example, let the function formula_9 be defined for all real numbers "x" as: and let us evaluate the following sequence of approximations of the integral: If we let "x" approach infinity, we get the following sequence: The limit of this infinite sequence of approximations is therefore the function formula_10. A function of two variables may be integrated over a closed interval containing the interval of integration only if one of the functions is differentiable at every point in the closed interval. The indefinite integral of a function of two variables over a closed interval containing the interval of integration is the limit of the sequence of indefinite integrals of the function of the function. The limit of an infinite sequence of indefinite integrals
