# Integral

lies to the right of one with a lower index. The values a and b, the end-points of the interval, are called the limits of integration of f. Integrals can also be defined if a > b: Reversing limits of integration. If a > b then define This, with a = b, implies: Integrals over intervals of length zero. If a is a real number then The first convention is necessary in consideration of taking integrals over subintervals of [un, b]; the second says that an integral taken over a degenerate interval, or a point, should be zero. One reason for the first convention is that the integrability of f on an interval [un, b] implies that f is integrable on any subinterval [c, d], but in particular integrals have the property that: Additivity of integration on intervals. If c is any element of [un, b], then With the first convention the resulting relation is then well-defined for any cyclic permutation of a, b, and c. Instead of viewing the above as conventions, one can also adopt the point of view that integration is performed on oriented manifolds only. If M is such an oriented m-dimensional manifold, and M' is the same manifold with opposed orientation and ω is an m-form, then one has (see below for integration of differential forms): Fundamental theorem of calculus Main article: Fundamental theorem of calculus The fundamental theorem of calculus is the statement that differentiation and integration are inverse operations: if a continuous function is first integrated and then differentiated, the original function is retrieved. An important consequence, sometimes called the second fundamental theorem of calculus, allows one to compute integrals by using an antiderivative of the function to be integrated. Statements of theorems Fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [un, b]. If F is defined for x in [un, b] by then F is continuous on [un, b]. If f is continuous at x in [un, b], then F is differentiable at x, and F ′(x) = f(x). Second fundamental theorem of calculus. Let f be a real-valued integrable function defined on a closed interval [un, b]. If F is a function such that F ′(x) = f(x) for all x in [un, b] (Es decir, F is an antiderivative of f), then Corollary. If f is a continuous function on [un, b], then f is integrable on [un, b], y F, defined by is an anti-derivative of f on [un, b]. Además, Extensions Improper integrals Main article: Improper integral File:Improper integral.svg The improper integral has unbounded intervals for both domain and range. Un "correcto" 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, por ejemplo (un,b], then again a limit may provide a finite result. Es decir, the improper integral is the limit of proper integrals as one endpoint of the interval of integration approaches either a specified real number, or ∞, or −∞. In more complicated cases, limits are required at both endpoints, or at interior points. Consider, por ejemplo, the function integrated from 0 to ∞ (shown right). At the lower bound, as x goes to 0 the function goes to ∞, and the upper bound is itself ∞, though the function goes to 0. Thus this is a doubly improper integral. Integrated, decir, De 1 Para 3, an ordinary Riemann sum suffices to produce a result of . To integrate from 1 to ∞, a Riemann sum is not possible. Sin embargo, any finite upper bound, say t (with t > 1), gives a well-defined result, . This has a finite limit as t goes to infinity, a saber . Semejantemente, the integral from 1⁄3 to 1 allows a Riemann sum as well, coincidentally again producing . Replacing 1⁄3 by an arbitrary positive value s (with s < 1) is equally safe, giving . This, too, has a finite limit as s goes to zero, namely . Combining the limits of the two fragments, the result of this improper integral is This process is not guaranteed success; a limit may fail to exist, or may be unbounded. For example, over the bounded interval 0 to 1 the integral of does not converge; and over the unbounded interval 1 to ∞ the integral of does not converge. It may also happen that an integrand is unbounded at an interior point, in which case the integral must be split at that point, and the limit integrals on both sides must exist and must be bounded. Thus But the similar integral cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see Cauchy principal value.) Multiple integration Main article: Multiple integral File:Volume under surface.png Double integral as volume under a surface. Integrals can be taken over regions other than intervals. In general, an integral over a set E of a function f is written: Here x need not be a real number, but can be another suitable quantity, for instance, a vector in R3. Fubini's theorem shows that such integrals can be rewritten as an iterated integral. In other words, the integral can be calculated by integrating one coordinate at a time. 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 which contains its domain. (The same volume can be obtained via the triple integral — the integral of a function in three variables — of the constant function f(x, y, z) = 1 over the above-mentioned region between the surface and the plane.) If the number of variables is higher, then the integral represents a hypervolume, a volume of a solid of more than three dimensions that cannot be graphed. For example, the volume of the cuboid of sides 4 × 6 × 5 may be obtained in two ways: By the double integral of the function f(x, y) = 5 calculated in the region D in the xy-plane which is the base of the cuboid. For example, if a rectangular base of such a cuboid is given via the xy inequalities 2 ≤ x ≤ 7, 4 ≤ y ≤ 9, our above double integral now reads From here, integration is conducted with respect to either x or y first; in this example, integration is first done with respect to x as the interval corresponding to x is the inner integral. Once the first integration is completed via the method or otherwise, the result is again integrated with respect to the other variable. The result will equate to the volume under the surface. By the triple integral of the constant function 1 calculated on the cuboid itself. Line integrals Main article: Line integral File:Line-Integral.gif A line integral sums together elements along a curve. 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). 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 multiplied by distance may be expressed (in terms of vector quantities) as: ; which is paralleled by the line integral: ; which sums up vector components along a continuous path, and thus finds the work done on an object moving through a field, such as an electric or gravitational field Surface integrals Main article: Surface integral File:Surface integral illustration.png The definition of surface integral relies on splitting the surface into small surface elements. A surface integral is a definite integral taken 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 function to be integrated may be a scalar field or a vector field. 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. Integrals of differential forms Main article: differential form A differential form is a mathematical concept in the fields of multivariable calculus, differential topology and tensors. The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan. We initially work in an open set in Rn. A 0-form is defined to be a smooth function f. When we integrate a function f over an m-dimensional subspace S of Rn, we write it as (The superscripts are indices, not exponents.) We can consider dx1 through dxn to be formal objects themselves, rather than tags appended to make integrals look like Riemann sums. Alternatively, we can view them as covectors, and thus a measure of "density" (hence integrable in a general sense). We call the dx1, …,dxn basic 1-forms. We define the wedge product, "∧", a bilinear "multiplication" operator on these elements, with the alternating property that for all indices a. Note that alternation along with linearity implies dxb∧dxa = −dxa∧dxb. This also ensures that the result of the wedge product has an orientation. We define the set of all these products to be basic 2-forms, and similarly we define the set of products of the form dxa∧dxb∧dxc to be basic 3-forms. A general k-form is then a weighted sum of basic k-forms, where the weights are the smooth functions f. Together these form a vector space with basic k-forms as the basis vectors, and 0-forms (smooth functions) as the field of scalars. The wedge product then extends to k-forms in the natural way. Over Rn at most n covectors can be linearly independent, thus a k-form with k > n will always be zero, by the alternating property. In addition to the wedge product, there is also the exterior derivative operator d. This operator maps k-forms to (k+1)-forms. For a k-form ω = f dxa over Rn, we define the action of d by: with extension to general k-forms occurring linearly. This more general approach allows for a more natural coordinate-free approach to integration on manifolds. It also allows for a natural generalisation of the fundamental theorem of calculus, called Stokes' theorem, which we may state as where ω is a general k-form, and ∂Ω denotes the boundary of the region Ω. Thus in the case that ω is a 0-form and Ω is a closed interval of the real line, this reduces to the fundamental theorem of calculus. In the case that ω is a 1-form and Ω is a 2-dimensional region in the plane, the theorem reduces to Green's theorem. Semejantemente, using 2-forms, and 3-forms and Hodge duality, we can arrive at Stokes' theorem and the divergence theorem. In this way we can see that differential forms provide a powerful unifying view of integration. Methods Computing integrals The most basic technique for computing definite integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this: Let f(x) be the function of x to be integrated over a given interval [un, b]. Find an antiderivative of f, Es decir, a function F such that F' = f on the interval. Entonces, by the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration, Note that the integral is not actually the antiderivative, but the fundamental theorem allows us to use antiderivatives to evaluate definite integrals. The difficult step is often finding an 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. Techniques include: Integration by substitution Integration by parts Changing the order of integration Integration by trigonometric substitution Integration by partial fractions Integration by reduction formulae Integration using parametric derivatives Integrating trigonometric products as complex exponentials Differentiation under the integral sign Contour Integration Even if these techniques fail, it may still be possible to evaluate a given integral. 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; por ejemplo, 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. Symbolic algorithms Main article: Symbolic integration Many problems in mathematics, física, 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 presents a special challenge in the development of such systems. 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. Por ejemplo, it is known that the antiderivatives of the functions exp ( x2), xx and sin x /x cannot be expressed in the closed form involving only rational and exponential functions, logarithm, trigonometric and inverse trigonometric functions, and the operations of multiplication and composition; en otras palabras, none of the three given functions is integrable in elementary functions. Differential Galois theory provides general criteria that allow one to determine whether the antiderivative of an elementary function is elementary. Desgraciadamente, it turns out that functions with closed expressions of antiderivatives are the exception rather than the rule. Por consiguiente, 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. En particular, it may be useful to have, in the set of antiderivatives, the special functions of physics (like the Legendre functions, the hypergeometric function, the Gamma function and so on). Extending the Risch-Norman algorithm so that it includes these functions is possible but challenging. Most humans are not able to integrate such general formulae, so in a sense computers are more skilled at integrating highly complicated formulae. Very complex formulae are unlikely to have closed-form antiderivatives, so how much of an advantage this presents is a philosophical question that is open for debate. Numerical quadrature Main article: numerical integration The integrals encountered in a basic calculus course are deliberately chosen for simplicity; those found in real applications are not always so accommodating. Some integrals cannot be found exactly, some require special functions which themselves are a challenge to compute, and others are so complex that finding the exact answer is too slow. This motivates the study and application of numerical methods for approximating integrals, which today use floating point arithmetic on digital electronic computers. Many of the ideas arose much earlier, for hand calculations; but the speed of general-purpose computers like the ENIAC created a need for improvements. The goals of numerical integration are accuracy, fiabilidad, efficiency, and generality. Sophisticated methods can vastly outperform a naive method by all four measures (Dahlquist & Björck 2008; Kahaner, Moler & Nash 1989; Stoer & Bulirsch 2002). Consider, por ejemplo, the integral which has the exact answer 94⁄25 = 3.76. (In ordinary practice the answer is not known in advance, so an important task — not explored here — is to decide when an approximation is good enough.) A “calculus book” approach divides the integration range into, decir, 16 equal pieces, and computes function values. Spaced function values x −2.00 −1.50 −1.00 −0.50 0.00 0.50 1.00 1.50 2.00 F(x) 2.22800 2.45663 2.67200 2.32475 0.64400 −0.92575 −0.94000 −0.16963 0.83600 x −1.75 −1.25 −0.75 −0.25 0.25 0.75 1.25 1.75 F(x) 2.33041 2.58562 2.62934 1.64019 −0.32444 −1.09159 −0.60387 0.31734 Archivo:Numerical quadrature 4up.png Numerical quadrature methods: ■ Rectangle, ■ Trapezoid, ■ Romberg, ■ Gauss Using the left end of each piece, the rectangle method sums 16 function values and multiplies by the step width, h, aquí 0.25, to get an approximate value of 3.94325 for the integral. The accuracy is not impressive, but calculus formally uses pieces of infinitesimal width, so initially this may seem little cause for concern. En efecto, repeatedly doubling the number of steps eventually produces an approximation of 3.76001. Sin embargo 218 pieces are required, a great computational expense for so little accuracy; and a reach for greater accuracy can force steps so small that arithmetic precision becomes an obstacle. A better approach replaces the horizontal tops of the rectangles with slanted tops touching the function at the ends of each piece. This trapezium rule is almost as easy to calculate; it sums all 17 function values, but weights the first and last by one half, and again multiplies by the step width. This immediately improves the approximation to 3.76925, which is noticeably more accurate. Además, solamente 210 pieces are needed to achieve 3.76000, substantially less computation than the rectangle method for comparable accuracy. Romberg's method builds on the trapezoid method to great effect. Primero, the step lengths are halved incrementally, giving trapezoid approximations denoted by T(h0), T(h1), y así, where hk+1 is half of hk. For each new step size, only half the new function values need to be computed; the others carry over from the previous size (as shown in the table above). But the really powerful idea is to interpolate a polynomial through the approximations, and extrapolate to T(0). With this method a numerically exact answer here requires only four pieces (five function values)! The Lagrange polynomial interpolating {hk,T(hk)}k=0…2 = {(4.00,6.128), (2.00,4.352), (1.00,3.908)} is 3.76+0.148h2, producing the extrapolated value 3.76 at h = 0. Gaussian quadrature often requires noticeably less work for superior accuracy. En este ejemplo, it can compute the function values at just two x positions, ±2⁄√3, then double each value and sum to get the numerically exact answer. The explanation for this dramatic success lies in error analysis, and a little luck. An n-point Gaussian method is exact for polynomials of degree up to 2n−1. The function in this example is a degree 3 polynomial, plus a term that cancels because the chosen endpoints are symmetric around zero. (Cancellation also benefits the Romberg method.) Shifting the range left a little, so the integral is from −2.25 to 1.75, removes the symmetry. No obstante, the trapezoid method is rather slow, the polynomial interpolation method of Romberg is acceptable, and the Gaussian method requires the least work — if the number of points is known in advance. As well, rational interpolation can use the same trapezoid evaluations as the Romberg method to greater effect. Quadrature method cost comparison Method Trapezoid Romberg Rational Gauss Points 1048577 257 129 36 Rel. Err. −5.3×10−13 −6.3×10−15 8.8×10−15 3.1×10−15 Value In practice, each method must use extra evaluations to ensure an error bound on an unknown function; this tends to offset some of the advantage of the pure Gaussian method, and motivates the popular Gauss–Kronrod quadrature formulas. Symmetry can still be exploited by splitting this integral into two ranges, from −2.25 to −1.75 (no symmetry), and from −1.75 to 1.75 (symmetry). More broadly, adaptive quadrature partitions a range into pieces based on function properties, so that data points are concentrated where they are needed most. This brief introduction omits higher-dimensional integrals (por ejemplo, area and volume calculations), where alternatives such as Monte Carlo integration have great importance. A calculus text is no substitute for numerical analysis, but the reverse is also true. Even the best adaptive numerical code sometimes requires a user to help with the more demanding integrals. Por ejemplo, improper integrals may require a change of variable or methods that can avoid infinite function values; and known properties like symmetry and periodicity may provide critical leverage. See also Lists of integrals - integrals of the most common functions. Multiple integral Antiderivative Numerical integration Integral equation Riemann integral Riemann-Stieltjes integral Henstock–Kurzweil integral Lebesgue integration Darboux integral Riemann sum Product integral References Apostol, Tom M. (1967), Cálculo, Para.. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), Wiley, ISBN 978-0-471-00005-1 Bourbaki, Nicolás (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1. In particular chapters III and IV. Burton, David M. (2005), The History of Mathematics: Una introducción (6ª ed.), McGraw-Hill, p. p. 359, ISBN 978-0-07-305189-5 Cajori, Florian (1929), A History Of Mathematical Notations Volume II, Open Court Publishing, páginas. 247–252, ISBN 978-0-486-67766-8, HTTP://www.archive.org/details/historyofmathema027671mbp Dahlquist, Germund; Björck, Åke (2008), "Capítulo 5: Numerical Integration", Numerical Methods in Scientific Computing, Volume I, Filadelfia: SIAM, HTTP://www.mai.liu.se/~akbjo/NMbook.html Folland, Gerald B. (1984), Real Analysis: Modern Techniques and Their Applications (1ed.), John Wiley & Sons, ISBN 978-0-471-80958-6 Fourier, Jean Baptiste Joseph (1822), Théorie analytique de la chaleur, Chez Firmin Didot, père et fils, p. §231, HTTP://libros.google.com/books?id=TDQJAAAAIAAJ Available in translation as Fourier, Joseph (1878), The analytical theory of heat, Hombre libre, Alexander (trans.), Cambridge University Press, páginas. páginas. 200–201, HTTP://www.archive.org/details/analyticaltheory00fourrich Heath, T. L., Ed. (2002), The Works of Archimedes, Dover, ISBN 978-0-486-42084-4, HTTP://www.archive.org/details/worksofarchimede029517mbp (Originally published by Cambridge University Press, 1897, based on J. L. Heiberg's Greek version.) Hildebrandt, T. H. (1953), "Integration in abstract spaces", Bulletin of the American Mathematical Society 59 (2): 111–139, ISSN 0273-0979, HTTP://projecteuclid.org/euclid.bams/1183517761 Kahaner, David; Moler, Cleve; Nash, Esteban (1989), "Capítulo 5: Numerical Quadrature", Numerical Methods and Software, Prentice Hall, ISBN 978-0-13-627258-8 Leibniz, Gottfried Wilhelm (1899), Gerhardt, Karl Immanuel, Ed., Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band, Berlín: Mayer & Müller, HTTP://name.umdl.umich.edu/AAX2762.0001.001 Miller, Jeff, Earliest Uses of Symbols of Calculus, Archived from the original on 1998-12-05, HTTP://web.archive.org/19981205110714/members.aol.com/jeff570/calculus.html, recuperado el 2007-06-02 O’Connor, J. J.; robertson, E. F. (1996), A history of the calculus, HTTP://www-history.mcs.st-andrews.ac.uk/HistTopics/The_rise_of_calculus.html, recuperado el 2007-07-09 Rudin, Walter (1987), "Capítulo 1: Abstract Integration", Real and Complex Analysis (International ed.), McGraw-Hill, ISBN 978-0-07-100276-9 Saks, Stanisław (1964), Theory of the integral (English translation by L. C. Joven. With two additional notes by Stefan Banach. Second revised ed.), Nueva York: Dover, HTTP://matwbn.icm.edu.pl/kstresc.php?tom=7&wyd=10&jez= Stoer, Josef; Bulirsch, Roland (2002), "Capítulo 3: Topics in Integration", Introduction to Numerical Analysis (3rd ed.), Salmer, ISBN 978-0-387-95452-3. W3C (2006), Arabic mathematical notation, HTTP://www.w3.org/TR/arabic-math/ ↑ Shea, Marilyn (Mayo 2007), Biography of Zu Chongzhi, Universidad de Maine, HTTP://hua.umf.maine.edu/China/astronomy/tianpage/0014ZuChongzhi9296bw.html, recuperado el 9 Enero 2009 Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison Wesley, páginas. 125–126, ISBN 978-0-321-16193-2 ↑ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163-174 [165] ↑ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163–174 [165–9 & 173–4] ↑ Bartle, Robert G. (1996). Return of the Riemann Integral. The American Mathematical Monthly 103: 625-632. External links The Integrator by Wolfram Research Function Calculator from WIMS Mathematical Assistant on Web online calculation of integrals, allows to integrate in small steps (includes also hints for next step which cover techniques like by parts, substitution, partial fractions, application of formulas and others, powered by Maxima (software)) Online books Keisler, H. Jerome, Elementary Calculus: An Approach Using Infinitesimals, University of Wisconsin Stroyan, K.D., A Brief Introduction to Infinitesimal Calculus, University of Iowa Mauch, sean, Sean's Applied Math Book, CIT, an online textbook that includes a complete introduction to calculus Crowell, Benjamín, Cálculo, Fullerton College, an online textbook Garrett, Paul, Notes on First-Year Calculus Hussain, Faraz, Understanding Calculus, an online textbook Kowalk, W.P., Integration Theory, University of Oldenburg. A new concept to an old problem. Online textbook Sloughter, Dan, Difference Equations to Differential Equations, an introduction to calculus Numerical Methods of Integration at Holistic Numerical Methods Institute P.S. Wang, Evaluation of Definite Integrals by Symbolic Manipulation (1972) - a cookbook of definite integral techniques This page uses Creative Commons Licensed content from Wikipedia (ver autores).

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