Applications of Integration
In mathematics and physics integration is a process of finding an area or volume by tracing out its boundary.. Integration is a fundamental tool of calculus and its various applications in science and engineering. Integration is a key concept in several mathematical disciplines, including differential calculus, complex analysis, topology and functional analysis. A: In mathematics, the integral of a function of a variable takes values in a function's range. When the function is evaluated at a particular value of its independent variable, the result of the integral is the value of the function at that point. The integrals of the elementary functions, such as the sine and cosine functions, are defined using the fundamental theorem of calculus, which in turn is based on the differential calculus. The elementary functions are used to define the more complicated functions by integration, which is the reverse process of differentiation. In calculus, the integrals of elementary functions which are defined by simple rules of the form are known as elementary functions. To take one example, the integral of the cosine function is For example, the integral of the secant function is In many applications, the integrals of any given function "f" of a real variable can be written in the form where "C" is a constant of integration known as the constant of integration. The constant of integration is often omitted in notation when the context is clear, as in The constant of integration may be omitted when the context is clear; for example, In the case of polynomial functions, the constant of integration is sometimes called the constant of integration or leading coefficient. For example, the integral of the polynomial function can be written as The constant of integration is an integer, and the integral of a polynomial function contains the nth degree term, or leading coefficient. In this case, the constant of integration is the leading coefficient. The remaining terms are known as the constant term. The Laplace transform and Fourier transform are inverse operations to differentiation and integration. The concept of the integral was developed by Isaac Newton in his "Principia", in which he defined the area under a curve as the limit of sums of infinitesimal areas of rectangles fitted snugly under the curve. Newton's method is the method of exhaustion, in which an area is approximated by successively adding up a series of rectangles whose widths are in a fixed ratio to the area sought. Here is an example with a rectangle of width 1 and length "x" and of height "y" (written in terms of the trigonometric functions): This is a simple example of a method of exhaustion, in which the width of each rectangle "varies in proportion to the area to be approximated". The method of limits was developed by Carl Friedrich Gauss and Jakob Bernoulli at the end of the 17th century. Here is an example of the method of limits, again with a rectangle of width "x" and length "y" and of height "z". The method of least squares is an example of the method of variation of variables. The basic idea is to vary one of the parameters of the problem, for example, the width of the rectangle. The approximate area of the rectangle is then calculated and compared with the exact area. The first step is to find the approximate area of the rectangle. This can be done by first finding the area of a smaller rectangle with the same height and width, but a smaller width. Then, the ratio of the area of this smaller rectangle to the area of the larger rectangle is used to find the width of the larger rectangle. The method of limits is used to find the approximate area of the larger rectangle. In the case of a curve, the concept of an integral can be extended to include areas under the graph of the function. For example, the area under the graph of the function is the integral of the function from a limit as "x" approaches 0 to infinity. The area under the curve between the limits 0 and 1 is the limit as "x" approaches 1 from 0, and is given by the definite integral of the function from 0 to 1. The area under the curve between the limits 1 and 1 is the limit as "x" approaches 1 from 1, and is given by the indefinite integral of the function from 1 to 1. The idea of integrating the area between a curve and the x-axis is also used to find the area between two curves. For example, the area between the graph of the function and the line "y" = "x" is given by the definite integral of the function from 0 to 1. The area between the graph of the function and the line "y" = "x" ? "a" is given by the indefinite integral of
