User:Cronholm144/Integral

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The integral between a and b of f(x) is the area between the curve y = f(x) and the x-axis in the interval [a, b].

In calculus, the integral of a function extends the concept of an ordinary sum. While an ordinary sum is taken over a discrete set of values, integration extends this concept to sums over continuous domains. The process of evaluating (or determining) an integral is known as integration. Integration is used to find the "total amount" of a property, whenever that property varies or is distributed in a known manner across a continuous domain. For example, instantaneous velocity changes moment to moment through the continuous domain of time. To sum up all the instantaneous velocities over a given interval of time, and hence obtain the total displacement that occurred, we evaluate the integral of the velocity over the given interval of time. Though this concept was the starting point for the development of integration theory by Newton and Leibniz, it has since been extended and newer definitions stress different aspects.

If a function has an integral, it is said to be integrable. The function for which the integral is calculated is called the integrand. The region over which a function is being integrated is called the domain of integration. In general the integrand may be a function of more than one variable, and the domain of integration may be an area, volume, a higher dimensional region, or even an abstract space that does not have a geometric structure in any usual sense. The simplest case, the integral of a real-valued function f of one real variable x on the interval [a, b], is denoted:

\int_a^b f(x)\,dx

The ∫ sign represents integration; a and b are the lower limit and upper limit of integration, defining the domain of integration; f(x) is the integrand; and dx is a notation for the variable of integration.

This form of integral may be identified with the signed area under the curve defined by the graph of f over the interval [a,b].

The term "integral" may also refer to antiderivatives. Though they are closely related through the fundamental theorem of calculus, the two notions are conceptually distinct. When one wants to clarify this distinction, an antiderivative is referred to as an indefinite integral (a function), while the integrals discussed in this article are termed definite integrals.

Informal discussion[edit]

Integrals generalize discrete sums to continuous sums. Suppose we had a list providing the population of every country in the world. It would be reasonable to ask about the total population of some region of the world; the total population of Africa, say. To find the total population of Africa we would sum up all the populations of countries that were located in Africa. Visualizing this, we could imagine the list depicted as a bar chart, one bar for each country, and each bar of unit width ("country" being the unit with respect to which the list was given). The total population of Africa can then be thought of as the area formed by all the bars associated with countries located in Africa. Since each country is discrete and distinct, and there are only finitely many countries in Africa, performing the sum, or finding the total area of bars in the chart, is a simple procedure. If, rather than populations of countries, we are given measurements of the instantaneous speed of a car through the continuous span of a 5 minute interval of time, we have some difficulties if we wish to sum up the speeds over the first 10 seconds. The first difficulty is that time is not discrete; there is no "smallest unit" of time that we may use as base unit (whereas countries provide the base unit in the discrete example). The second difficulty, linked with the first, is there are not finitely many measurements of instantaneous speed which we would have to sum; since time is continuous any interval of time will contain an infinite number of "instants".

Visualizing the problem, we see that we can express the list instantaneous speed measurements as a function of time, and to sum up the instantaneous speeds over the first 10 seconds is to find the area beneath the graph of that function from 0 to 10 seconds. We can, therefore, approximate the area as follows: choose some unit of time, say 1 second, as a base unit -- it will not be exact, since there are infinitely many instants in even a single second, but it will suffice for approximation -- and construct rectangles of width 1 second and height approximating the value of the function over 1 second time span of the rectangle; we then have a finite number of discrete bars, which we can sum the area of in the usual way. In doing this we do, of course, have to choose a particular time in each one second span which has associated with it the instantaneous speed we want to use to give the height of the rectangle. If we choose a smaller base unit, say half a second, and pick out a particular time for each half second interval to provide the height for each rectangle, we get a slightly better approximation of the area. Thus by taking progressively smaller and smaller base units we get closer and closer to determining the area under the curve. Taking the limit of the area over ever finer divisions of time we can evaluate the area under the curve. This process is essentially Riemann integration.

The process so far described is similar to the method of exhaustion used by the Greeks. One of the major developments of Calculus in the 17th century was relating integrals to derivatives via the fundamental theorem of calculus; this made the calculation of integrals much easier, and allowed a much broader range of integrals to be evaluated. To see the connection in the example above, consider that each instantaneous speed is a measure of the rate of change of (positional) displacement (with respect to time); thus each rectangle in our approximation gives the total (positional) displacement, since its area is the rate of change of displacement (height) multiplied by the length of time over which that rate of displacement occurs (width). That is, the area under the curve is the total displacement that occurs in the first 10 seconds. Conversely, if we were to plot total displacement against time then the instantaneous slope of the resulting curve (the derivative) would give the instantaneous speed.

Formal definition[edit]

There are many ways of formally defining an integral, not all of which are equivalent. The differences exist mostly to deal with differing special cases which may not be integrable under other definitions, but also occasionally for pedagogical reasons. The most commonly used definitions of integral are Riemann integrals, which provided the first rigorous formal definition of an integral, and Lebesgue integrals.

Riemann Integral[edit]

Main article: Riemann integral
A sequence of Riemann sums. The numbers in the upper right are the areas of the gray rectangles. They converge to the integral of the function.

The formal definition of the Riemann integral is described in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval of the real line; a tagged partition of [a,b] is a finite sequence

 a = x_0 \le t_1 \le x_1 \le t_2 \le x_2 \le \cdots \le x_{n-1} \le t_n \le x_n = b . \,\!

This partitions the interval [a,b] into i sub-intervals [xi−1, xi], each of which is "tagged" with a distinguished point ti ∈ [xi−1, xi]. Let Δi = xixi−1 be the width of sub-interval i; then the mesh of such a tagged partition is maxi=1…n Δi; that is, the width of the largest sub-interval formed by the partition. A Riemann sum of a function f with respect to such a tagged partition is defined as

\sum_{i=1}^{n}f(t_i)\Delta_i;

thus each term of the sum is the area of rectangle with the same width as the sub-interval, and height equal to the function value at the distinguished point of the given sub-interval. We say that the Riemann integral of a function f over the interval [a,b] is equal to S if:

For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have
\left|\sum_{i=1}^{n}f(t_i)\Delta_i - S\right| < \epsilon.

Lebesgue Integral[edit]

Main article: Lebesgue integral

The Riemann definition of the integral has the difficulty that it is not defined for a wide range of (discontinuous) functions that proved to be of increasing interest and importance. This lead to a more technical definition, the Lebesgue integral, under which a much larger number of functions are integrable (Rudin 1987). According to Folland (1984, p. 56), "to compute the Riemann integral of f, one partitions the domain [a,b] into subintervals", while in the Lebesgue integral, "one is in effect partitioning the range of f".

The definition of the Lebesgue integral is grounded in measure theory. Lebesgue integrals are defined in terms of a measure μ (essentially a function that assigns to each "measurable" subset a number interpreted as its "size") and indicator functions. The case directly generalising the Riemann integral is the one where the measure μ is the Lebesgue measure, i.e. the measure which to an interval [a,b] associates its length ba.

In the most common approach, one first defines the integral of the indicator function of a measurable set S by:

\int 1_S d\mu = \mu(S).

This extends by linearity to measurable simple functions, s, which attain only a finite number, n, of distinct non-negative values:

\begin{align}
 \int s \, d\mu &{}= \int\left(\sum_{i=1}^{n} a_i 1_{S_i}\right) d\mu \\
  &{}= \sum_{i=1}^{n} a_i\int 1_{S_i} \, d\mu \\
  &{}= \sum_{i=1}^{n} a_i \, \mu(S_i)
\end{align}

(where the image of Si under the simple function s is the constant value ai). Thus if E is a measurable set one defines

 \int_E s \, d\mu = \sum_{i=1}^{n} a_i \, \mu(S_i \cap E) .

Then for any non-negative measurable function f one defines

\int_E f \, d\mu = \sup\left\{\int_E s \, d\mu\, \colon 0 \leq s\leq f\text{ and } s\text{ is a simple function}\right\};

that is, the integral of f is set to be the supremum of all the integrals of simple functions that are less than or equal to f. A general measurable function f, is split into its positive and negative values by defining

\begin{align}
 f^+(x) &{}= \begin{cases}
               f(x), & \text{if } f(x) > 0 \\
               0, & \text{otherwise}
             \end{cases} \\
 f^-(x) &{}= \begin{cases}
               -f(x), & \text{if } f(x) < 0 \\
               0, & \text{otherwise}
             \end{cases}
\end{align}

Finally, f is Lebesgue integrable if

\int_E |f| \, d\mu < \infty , \,\!

and then the integral is defined by

\int_E f \, d\mu = \int_E f^+ \, d\mu - \int_E f^- \, d\mu . \,\!

When the measure space on which the functions are defined is also a locally compact topological space (as is the case with the real numbers R), measures compatible with the topology in a suitable sense (Radon measures, of which the Lebesgue measure is an example) and integral with respect to them can be defined differently, starting from the integrals of continuous functions with compact support. More precisely, the compactly supported functions form a vector space that carries a natural topology, and a (Radon) measure can be defined as any continuous linear functional on this space; the value of a measure at a compactly supported function is then also by definition the integral of the function. One then proceeds to expand the measure (the integral) to more general functions by continuity, and defines the measure of a set as the integral of its indicator function. This is the approach taken by Bourbaki (2004) and a certain number of other authors. For details see Radon measures.

Other Integrals[edit]

Although the Riemann and Lebesgue integrals are the most important definitions of the integral, a number of others exist, including:

History[edit]

Main article: History of calculus

Pre-calculus integration[edit]

Integration can be traced back as far Egypt, circa 1800 BC, with the Moscow Mathematical Papyrus demonstrating knowledge of a formula for the volume of a pyramidal frustrum. The first documented systematic technique capable of determining integrals is the method of exhaustion of Eudoxus (circa 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of shapes for which the area or volume was known. This method was further developed and employed by Archimedes and used to calculate areas for parabolas and an approximation to the area of a circle. Similar methods were independently developed in China around the 3rd Century AD by Liu Hui, who used it to find the area of the circle. This method was later used by Zu Chongzhi to find the volume of a sphere.[1]

Significant advances on techniques such as the method of exhaustion did not begin to appear until the 16th Century AD. At this time the work of Cavalieri with his method of indivisibles, and work by Fermat, began to lay the foundations of modern calculus. Further steps were made in the early 17th Century by Barrow and Torricelli, who provided the first hints of a connection between integration and differentiation.

Newton and Leibniz[edit]

The major advance in integration came in the 17th Century with the independent discovery of the fundamental theorem of calculus by Newton and Leibniz. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern Calculus, whose notation for integrals is drawn directly from the work of Leibniz.

Formalising integrals[edit]

While Newton and Leibniz provided systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked infinitesimals as "the ghosts of departed quantity". Calculus acquired a firmer footing with the development of limits and was given a suitable foundation by Cauchy in the first half of the 19th century. Integration was first rigorously formalised, using limits, by Riemann. Although all continuous functions on a closed and bounded interval are Riemann integrable, subsequently more general functions were considered, to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory. Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed.

Notation[edit]

Isaac Newton used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with \dot{x} or x'\,\!, which Newton used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.

The modern notation for the indefinite integral was introduced by Gottfried Leibniz in 1675 (Burton 1988, p. 359)(Leibniz 1899, p. 154). He derived the integral symbol, "∫", from an elongated letter S, standing for summa (Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by Joseph Fourier in Mémoires of the French Academy around 1819–20, reprinted in his book of 1822 (Cajori 1929, pp. 249–250)(Fourier 1822, §231). In Arabic which is written from right to left, an inverted integral symbol Signed'IntegracióArabic.png is used (W3C 2006).

Forms and applications[edit]

Computing integrals[edit]

The most basic technique for computing integrals of one real variable is based on the fundamental theorem of calculus. It proceeds like this:

  1. Choose a function f(x) and an interval [a, b].
  2. Find an antiderivative of f, that is, a function F such that F' = f.
  3. By the fundamental theorem of calculus, provided the integrand and integral have no singularities on the path of integration,
    \int_a^b f(x)\,dx = F(b)-F(a).
  4. Therefore the value of the integral is F(b) − F(a).

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:

Even if these techniques fail, it may still be possible to evaluate a given integral. The next most common technique is residue calculus, whilst for nonelementary integrals Taylor series can sometimes be used to find the antiderivative. 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.

Improper integrals[edit]

Main article: Improper integral
The improper integral
\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pi
has unbounded intervals for both domain and range.

A "proper" Riemann integral assumes the integrand is finite and defined 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 the Riemann integral 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.

\int_{a}^{\infty} f(x)dx = \lim_{b \to \infty} \int_{a}^{b} f(x)dx

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.

\int_{a}^{b} f(x)dx = \lim_{\epsilon \to 0} \int_{a+\epsilon}^{b} f(x)dx

That is, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.

Consider, for example, the function 1/((x+1)√x) 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, say, from 1 to 3, an ordinary Riemann sum suffices to produce a result of π/6. To integrate from 1 to ∞, a Riemann sum is not possible. However, any finite upper bound, say t (with t > 1), gives a well-defined result, π/2 − 2 arctan(1/√t). This has a finite limit as t goes to infinity, namely π/2. Similarly, the integral from 13 to 1 allows a Riemann sum as well, coincidentally again producing π/6. Replacing 13 by an arbitrary positive value s (with s < 1) is equally safe, giving −π/2 + 2 arctan(1/√s). This, too, has a finite limit as s goes to zero, namely π/2. Combining the limits of the two fragments, the result of this improper integral is

\begin{align}
 \int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} &{} = \lim_{s \to 0} \int_{s}^{1} \frac{dx}{(x+1)\sqrt{x}}
   + \lim_{t \to \infty} \int_{1}^{t} \frac{dx}{(x+1)\sqrt{x}} \\
  &{} = \lim_{s \to 0} \left( -\frac{\pi}{2} + 2 \arctan(1/\sqrt{s}) \right)
   + \lim_{t \to \infty} \left( \frac{\pi}{2} - 2 \arctan(1/\sqrt{t}) \right) \\
  &{} = \frac{\pi}{2} + \frac{\pi}{2} \\
  &{} = \pi .
\end{align}

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 1/x2 does not converge; and over the unbounded interval 1 to ∞ the integral of 1/√x 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

\begin{align}
 \int_{-1}^{1} \frac{dx}{\sqrt[3]{x^2}} &{} = \lim_{s \to 0} \int_{-1}^{-s} \frac{dx}{\sqrt[3]{x^2}}
   + \lim_{t \to 0} \int_{t}^{1} \frac{dx}{\sqrt[3]{x^2}} \\
  &{} = \lim_{s \to 0} 3(1-\sqrt[3]{s}) + \lim_{t \to 0} 3(1-\sqrt[3]{t}) \\
  &{} = 3 + 3 \\
  &{} = 6.
\end{align}

But the similar integral

 \int_{-1}^{1} \frac{dx}{x} \,\!

cannot be assigned a value in this way, as the integrals above and below zero do not independently converge. (However, see Cauchy principal value.)

Numerical quadrature[edit]

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, reliability, efficiency, and generality. Sophisticated methods can vastly outperform a naive method by all four measures (Dahlquist & Björck forthcoming)(Kahaner, Moler & Nash 1989)(Stoer & Bulirsch 2002). Consider, for example, the integral

 \int_{-2}^{2} \tfrac15 \left( \tfrac{1}{100}(322 + 3 x (98 + x (37 + x))) - 24 \frac{x}{1+x^2} \right) dx ,

which has the exact answer 9425 = 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, say, 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
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, here 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. Indeed, repeatedly doubling the number of steps eventually produces an approximation of 3.76001. However 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 trapezoidal 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. Furthermore, only 210 pieces are needed to achieve 3.76000, substantially less computation than the rectangle method for comparable accuracy.

The Romberg method builds on the trapezoid method to great effect. First, the step lengths are halved incrementally, giving trapezoid approximations denoted by T(h0), T(h1), and so on, 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. 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 puts all these to shame. In this example, it computes the function values at just two x positions, ±2√3, then doubles each value and sums 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. Nevertheless, 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 33
Rel. Err. −5.3×10−13 −6.3×10−15 8.8×10−15 −5.6×10−14
Value \textstyle \int_{-2.25}^{1.75} f(x)\,dx = 4.1639019006585897075\ldots

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 hybrid. 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 (for example, 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. For example, 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.

Symbolic integration[edit]

Main article: Symbolic integration

Many professionals, educators, and students now use computer algebra systems to make difficult (or simply tedious) algebra and calculus problems easier. The design of such a computer algebra system is nontrivial as systematic methods of antidifferentiation are difficult to formulate, although in many cases a definite integral can be computed without finding an antiderivative.

One difficulty in computing definite integrals is that it is not always possible to find "explicit formulae" for antiderivatives. For instance, there is a (nontrivial) proof that there is no elementary function (e.g., involving sin, cos, exp, polynomials, roots and so on) whose derivative is xx. As such, computerized algebra systems have no hope of being able to find an antiderivative for this particular function. Unfortunately, functions that have nice antiderivatives are the exception. If one writes a large random expression involving exponentials and polynomials, the odds are almost nil that it will have a "nice" antiderivative. (This statement can be made formal, but it is difficult to do so.)

One of the difficulties is to decide what set of functions to use as building blocks for antiderivatives. Usually, we need a set of antiderivatives closed under, say, multiplication and composition. This set of antiderivatives should also include polynomials, perhaps quotients, exponentials, logarithms, sines and cosines. The Risch-Norman algorithm is able to compute any integral of such a shape; that is, if the antiderivative involves polynomials, sines, cosines, etc..., the Risch-Norman algorithm will be able to compute it. Extended versions of this algorithm are implemented in Mathematica and the Maple computer algebra system.

Some special integrands occur often enough to warrant special study. In 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. On the other hand, very complex formulae are unlikely to have closed-form antiderivatives, so this advantage is dubious.

Multiple integration[edit]

Main article: Multiple integral

Integrals can be taken over regions other than intervals. In general, an integral over a set E of a function f is written:

\int_E f(x) \, dx

Here x need not be a real number, but can be other suitable algebraic quantities. 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. (Note that 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, one will calculate "hypervolumes" (volumes of solid of more than three dimensions) that cannot be graphed.

Multiple integral as volume under a surface.

For example, the volume of the parallelepiped of sides 4×6×5 may be obtained in two ways:

  • By the double integral
\iint_D 5 \ dx\, dy
of the function f(x, y) = 5 calculated in the region D in the xy-plane which is the base of the parallelepiped.
  • By the triple integral
\iiint_\mathrm{parallelepiped} 1 \, dx\, dy\, dz
of the constant function 1 calculated on the parallelepiped itself.

Because it is impossible to calculate the antiderivative of a function of more than one variable, indefinite multiple integrals do not exist so they are all definite integrals.

Line and surface integrals[edit]

The integrals discussed so far are constrained to domains of integration that are "straight" or "flat": intervals of a straight line, regions in a flat plane, a volume in a "flat" 3-space, etc. The concept of an integral can be extended to more general domains of integration, however, such as curved lines and surfaces. Such integrals are known as line integrals and surface integrals respectively. Both line integrals and surface integrals are of importance in physics, often when dealing with vector fields

Line integrals[edit]

Main article: Line integral
A line integral sums together elements along a curve.

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:

W=\vec F\cdot\vec d;

which is paralleled by the line integral:

W=\int_C \vec F\cdot d\vec s;

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[edit]

Main article: Surface integral
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:

\int_S {\mathbf v}\cdot \,d{\mathbf {S}}.

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.

Integration of Differential Forms[edit]

A differential form is a mathematical concept in the fields of multivariate 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

\int_S f\,dx^1 \cdots dx^m.

(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

 dx^a \wedge dx^a = 0 \,\!

for all indices a. Note that alternation along with linearity implies dxbdxa = −dxadxb. 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 dxadxbdxc 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:

{\bold d}{\omega} = \sum_{i=1}^n \frac{\partial f}{\partial x_i} dx^i \wedge dx^a.

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 Stoke's theorem, which we may state as

\int_{\Omega} {\bold d}\omega = \int_{\partial\Omega} \omega \,\!

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. Similarly, using 2-forms, and 3-forms and Hodge duality, we can arrive at Stoke's theorem and the divergence theorem. In this way we can see that differential forms provide a powerful unifying view of integration.

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