Davis , Philip Rabinowitz. Methods of Numerical Integration, Second Edition describes the theoretical and practical aspects of major methods of numerical integration. Numerical integration is the study of how the numerical value of an integral can be found. This book contains six chapters and begins with a discussion of the basic principles and limitations of numerical integration.
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The succeeding chapters present the approximate integration rules and formulas over finite and infinite intervals. These topics are followed by a review of error analysis and estimation, as well as the application of functional analysis to numerical integration. A chapter describes the approximate integration in two or more dimensions. Then, F is continuous on [ a , b ] , differentiable on the open interval a , b , and. Let f be a real-valued function defined on a closed interval [ a , b ] that admits an antiderivative F on [ a , b ].
That is, f and F are functions such that for all x in [ a , b ] ,. If f is integrable on [ a , b ] then. The second fundamental theorem allows many integrals to be calculated explicitly. For example, to calculate the integral.
Then the value of the integral in question is. Tables of this and similar antiderivatives can be used to calculate integrals explicitly, in much the same way that tables of derivatives can be used. A "proper" Riemann integral assumes the integrand is defined and finite on a closed and bounded interval, bracketed by the limits of integration. An improper integral occurs when one or more of these conditions is not satisfied.
In some cases such integrals may be defined by considering the limit of a sequence of proper Riemann integrals on progressively larger intervals. If the interval is unbounded, for instance at its upper end, then the improper integral is the limit as that endpoint goes to infinity. If the integrand is only defined or finite on a half-open interval, for instance a , b ] , then again a limit may provide a finite result. In more complicated cases, limits are required at both endpoints, or at interior points. Just as the definite integral of a positive function of one variable represents the area of the region between the graph of the function and the x -axis, the double integral of a positive function of two variables represents the volume of the region between the surface defined by the function and the plane that contains its domain.
Under suitable conditions e. This reduces the problem of computing a double integral to computing one-dimensional integrals. Because of this, another notation for the integral over R uses a double integral sign:. Integration over more general domains is possible. The concept of an integral can be extended to more general domains of integration, such as curved lines and surfaces.
Such integrals are known as line integrals and surface integrals respectively. These have important applications in physics, as when dealing with vector fields. A line integral sometimes called a path integral is an integral where the function to be integrated is evaluated along a curve.
Various different line integrals are in use. In the case of a closed curve it is also called a contour integral. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve.
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This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that work is equal to force , F , multiplied by displacement, s , may be expressed in terms of vector quantities as:. This gives the line integral. A surface integral generalizes double integrals to integration over a surface which may be a curved set in space ; it can be thought of as the double integral analog of the line integral.
The value of the surface integral is the sum of the field at all points on the surface. This can be achieved by splitting the surface into surface elements, which provide the partitioning for Riemann sums.
For an example of applications of surface integrals, consider a vector field v on a surface S ; that is, for each point x in S , v x is a vector. Imagine that we have a fluid flowing through S , such that v x determines the velocity of the fluid at x. The flux is defined as the quantity of fluid flowing through S in unit amount of time. To find the flux, we need to take the dot product of v with the unit surface normal to S at each point, which will give us a scalar field, which we integrate over the surface:. The fluid flux in this example may be from a physical fluid such as water or air, or from electrical or magnetic flux.
Thus surface integrals have applications in physics, particularly with the classical theory of electromagnetism. In complex analysis , the integrand is a complex-valued function of a complex variable z instead of a real function of a real variable x.
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This is known as a contour integral. A differential form is a mathematical concept in the fields of multivariable calculus , differential topology , and tensors. Differential forms are organized by degree. For example, a one-form is a weighted sum of the differentials of the coordinates, such as:. A differential one-form can be integrated over an oriented path, and the resulting integral is just another way of writing a line integral. Here the basic differentials dx , dy , dz measure infinitesimal oriented lengths parallel to the three coordinate axes.
Unlike the cross product, and the three-dimensional vector calculus, the wedge product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds curves, surfaces, and their higher-dimensional analogs. The exterior derivative plays the role of the gradient and curl of vector calculus, and Stokes' theorem simultaneously generalizes the three theorems of vector calculus: the divergence theorem , Green's theorem , and the Kelvin-Stokes theorem. The discrete equivalent of integration is summation.
Summations and integrals can be put on the same foundations using the theory of Lebesgue integrals or time scale calculus. The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus. Let f x be the function of x to be integrated over a given interval [ a , b ]. Provided the integrand and integral have no singularities on the path of integration, by the fundamental theorem of calculus,.
The integral is not actually the antiderivative, but the fundamental theorem provides a way to use antiderivatives to evaluate definite integrals. The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. More often, it is necessary to use one of the many techniques that have been developed to evaluate integrals. Most of these techniques rewrite one integral as a different one which is hopefully more tractable.
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Techniques include:. Alternative methods exist to compute more complex integrals. Many nonelementary integrals can be expanded in a Taylor series and integrated term by term. Occasionally, the resulting infinite series can be summed analytically. The method of convolution using Meijer G-functions can also be used, assuming that the integrand can be written as a product of Meijer G-functions.
There are also many less common ways of calculating definite integrals; for instance, Parseval's identity can be used to transform an integral over a rectangular region into an infinite sum. Occasionally, an integral can be evaluated by a trick; for an example of this, see Gaussian integral. Computations of volumes of solids of revolution can usually be done with disk integration or shell integration. Specific results which have been worked out by various techniques are collected in the list of integrals.
Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive tables of integrals have been compiled and published over the years for this purpose. With the spread of computers, many professionals, educators, and students have turned to computer algebra systems that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration has been one of the motivations for the development of the first such systems, like Macsyma.
A major mathematical difficulty in symbolic integration is that in many cases, a closed formula for the antiderivative of a rather simple-looking function does not exist. The Risch algorithm provides a general criterion to determine whether the antiderivative of an elementary function is elementary, and, if it is, to compute it. Unfortunately, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule.
Consequently, computerized algebra systems have no hope of being able to find an antiderivative for a randomly constructed elementary function. On the positive side, if the 'building blocks' for antiderivatives are fixed in advance, it may be still be possible to decide whether the antiderivative of a given function can be expressed using these blocks and operations of multiplication and composition, and to find the symbolic answer whenever it exists.
The Risch algorithm , implemented in Mathematica and other computer algebra systems , does just that for functions and antiderivatives built from rational functions, radicals , logarithm, and exponential functions. Some special integrands occur often enough to warrant special study. Extending the Risch's algorithm to include such functions is possible but challenging and has been an active research subject.