This article is about the concept of definite integrals in calculus. For the indefinite integral, see antiderivative. For the set of numbers, see integer. For other uses, see Integral (disambiguation).
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a functionf of a realvariablex and an interval[a, b] of the real line, the definite integral
is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total.
Roughly speaking, the operation of integration is the reverse of differentiation. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In this case, it is called an indefinite integral and is written:
The integrals discussed in this article are those termed definite integrals. It is the fundamental theorem of calculus that connects differentiation with the definite integral: if f is a continuous real-valued function defined on a closed interval[a, b], then, once an antiderivative F of f is known, the definite integral of f over that interval is given by
The principles of integration were formulated independently by Isaac Newton and Gottfried Leibniz in the late 17th century, who thought of the integral as an infinite sum of rectangles of infinitesimal width. Bernhard Riemann gave a rigorous mathematical definition of integrals. It is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into thin vertical slabs. Beginning in the nineteenth century, more sophisticated notions of integrals began to appear, where the type of the function as well as the domain over which the integration is performed has been generalised. A line integral is defined for functions of two or three variables, and the interval of integration [a, b] is replaced by a certain curve connecting two points on the plane or in the space. In a surface integral, the curve is replaced by a piece of a surface in the three-dimensional space.
See also: History of calculus
The first documented systematic technique capable of determining integrals is the method of exhaustion of the ancient Greek astronomer Eudoxus (ca. 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known. This method was further developed and employed by Archimedes in the 3rd century BC and used to calculate areas for parabolas and an approximation to the area of a circle.
A similar method was 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 in the 5th century by Chinese father-and-son mathematicians Zu Chongzhi and Zu Geng to find the volume of a sphere (Shea 2007; Katz 2004, pp. 125–126).
The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of Cavalieri with his method of Indivisibles, and work by Fermat, began to lay the foundations of modern calculus, with Cavalieri computing the integrals of xn up to degree n = 9 in Cavalieri's quadrature formula. 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. Barrow provided the first proof of the fundamental theorem of calculus. Wallis generalized Cavalieri's method, computing integrals of x to a general power, including negative powers and fractional powers.
Newton and Leibniz
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.
While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities". Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann. Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield of real analysis). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the standard part of an infinite Riemann sum, based on the hyperreal number system.
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 .x or x′, which are 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 adapted the integral symbol, ∫, from the letter ſ (long s), standing for summa (written as 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).
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Integrals are used extensively in many areas of mathematics as well as in many other areas that rely on mathematics.
For example, in probability theory, integrals are used to determine the probability of some random variable falling within a certain range. Moreover, the integral under an entire probability density function must equal 1, which provides a test of whether a function with no negative values could be a density function or not.
Integrals can be used for computing the area of a two-dimensional region that has a curved boundary, as well as computing the volume of a three-dimensional object that has a curved boundary.
Integrals are also used in physics, in areas like kinematics to find quantities like displacement, time, and velocity. For example, in rectilinear motion, the displacement of an object over the time interval is given by:
where is the velocity expressed as a function of time. The work done by a force (given as a function of position) from an initial position to a final position is:
Terminology and notation
The integral with respect to x of a real-valued functionf(x) of a real variable x on the interval [a, b] is written as
The integral sign ∫ represents integration. The symbol dx, called the differential of the variable x, indicates that the variable of integration is x. The function f(x) to be integrated is called the integrand. The symbol dx is separated from the integrand by a space (as shown). If a function has an integral, it is said to be integrable. The points a and b are called the limits of the integral. An integral where the limits are specified is called a definite integral. The integral is said to be over the interval [a, b].
If the integral goes from a finite value a to the upper limit infinity, the integral expresses the limit of the integral from a to a value b as b goes to infinity. If the value of the integral gets closer and closer to a finite value, the integral is said to converge to that value. If not, the integral is said to diverge.
When the limits are omitted, as in
the integral is called an indefinite integral, which represents a class of functions (the antiderivative) whose derivative is the integrand. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. Occasionally, limits of integration are omitted for definite integrals when the same limits occur repeatedly in a particular context. Usually, the author will make this convention clear at the beginning of the relevant text.
There are several extensions of the notation for integrals to encompass integration on unbounded domains and/or in multiple dimensions (see later sections of this article).
Meaning of the symbol dx
Historically, the symbol dx was taken to represent an infinitesimally "small piece" of the independent variable x to be multiplied by the integrand and summed up in an infinite sense. While this notion is still heuristically useful, later mathematicians have deemed infinitesimal quantities to be untenable from the standpoint of the real number system. In introductory calculus, the expression dx is therefore not assigned an independent meaning; instead, it is viewed as part of the symbol for integration and serves as its delimiter on the right side of the expression being integrated.
In more sophisticated contexts, dx can have its own significance, the meaning of which depending on the particular area of mathematics being discussed. When used in one of these ways, the original Leibnitz notation is co-opted to apply to a generalization of the original definition of the integral. Some common interpretations of dx include: an integrator function in Riemann-Stieltjes integration (indicated by dα(x) in general), a measure in Lebesgue theory (indicated by dμ in general), or a differential form in exterior calculus (indicated by in general). In the last case, even the letter d has an independent meaning — as the exterior derivative operator on differential forms.
Conversely, in advanced settings, it is not uncommon to leave out dx when only the simple Riemann integral is being used, or the exact type of integral is immaterial. For instance, one might write to express the linearity of the integral, a property shared by the Riemann integral and all generalizations thereof.
In modern Arabic mathematical notation, a reflected integral symbol is used instead of the symbol ∫, since the Arabic script and mathematical expressions go right to left. Some authors, particularly of European origin, use an upright "d" to indicate the variable of integration (i.e., dx instead of dx), since properly speaking, "d" is not a variable. Also, the symbol dx is not always placed after f(x), as for instance in
- or .
In the first expression, the differential is treated as an infinitesimal "multiplicative" factor, formally following a "commutative property" when "multiplied" by the expression 3/(x2+1). In the second expression, showing the differentials first highlights and clarifies the variables that are being integrated with respect to, a practice particularly popular with physicists.
Interpretations of the integral
Integrals appear in many practical situations. If a swimming pool is rectangular with a flat bottom, then from its length, width, and depth we can easily determine the volume of water it can contain (to fill it), the area of its surface (to cover it), and the length of its edge (to rope it). But if it is oval with a rounded bottom, all of these quantities call for integrals. Practical approximations may suffice for such trivial examples, but precision engineering (of any discipline) requires exact and rigorous values for these elements.
To start off, consider the curve y = f(x) between x = 0 and x = 1 with f(x) = √x (see figure). We ask:
- What is the area under the function f, in the interval from 0 to 1?
and call this (yet unknown) area the (definite) integral of f. The notation for this integral will be
As a first approximation, look at the unit square given by the sides x = 0 to x = 1 and y = f(0) = 0 and y = f(1) = 1. Its area is exactly 1. Actually, the true value of the integral must be somewhat less than 1. Decreasing the width of the approximation rectangles and increasing the number of rectangles gives a better result; so cross the interval in five steps, using the approximation points 0, 1/5, 2/5, and so on to 1. Fit a box for each step using the right end height of each curve piece, thus √1/5, √2/5, and so on to √1 = 1. Summing the areas of these rectangles, we get a better approximation for the sought integral, namely
We are taking a sum of finitely many function values of f, multiplied with the differences of two subsequent approximation points. We can easily see that the approximation is still too large. Using more steps produces a closer approximation, but will always be too high and will never be exact. Alternatively, replacing these subintervals by ones with the left end height of each piece, we will get an approximation that is too low: for example, with twelve such subintervals we will get an approximate value for the area of 0.6203.
The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely many fine, or infinitesimal steps. When this transition is completed in the above example, it turns out that the area under the curve within the stated bounds is 2/3.
conceives the integral as a weighted sum, denoted by the elongated s, of function values, f(x), multiplied by infinitesimal step widths, the so-called differentials, denoted by dx.
Historically, after the failure of early efforts to rigorously interpret infinitesimals, Riemann formally defined integrals as a limit of weighted sums, so that the dx suggested the limit of a difference (namely, the interval width). Shortcomings of Riemann's dependence on intervals and continuity motivated newer definitions, especially the Lebesgue integral, which is founded on an ability to extend the idea of "measure" in much more flexible ways. Thus the notation
refers to a weighted sum in which the function values are partitioned, with μ measuring the weight to be assigned to each value. Here A denotes the region of integration.
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 and Lebesgue integrals.
Main article: Riemann integral
The Riemann integral is defined 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; then a tagged partition of [a, b] is a finite sequence
This partitions the interval [a, b] into n sub-intervals [xi−1, xi] indexed by i, each of which is "tagged" with a distinguished point ti ∈ [xi−1, xi]. A Riemann sum of a function f with respect to such a tagged partition is defined as
thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let Δi = xi−xi−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, maxi=1...n Δi. 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
When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.
Main article: Lebesgue integration
It is often of interest, both in theory and applications, to be able to pass to the limit under the integral. For instance, a sequence of functions can frequently be constructed that approximate, in a suitable sense, the solution to a problem. Then the integral of the solution function should be the limit of the integrals of the approximations. However, many functions that can be obtained as limits are not Riemann-integrable, and so such limit theorems do not hold with the Riemann integral. Therefore, it is of great importance to have a definition of the integral that allows a wider class of functions to be integrated (Rudin 1987).
Such an integral is the Lebesgue integral, that exploits the following fact to enlarge the class of integrable functions: if the values of a function are rearranged over the domain, the integral of a function should remain the same. Thus Henri Lebesgue introduced the integral bearing his name, explaining this integral thus in a letter to Paul Montel:
Darboux upper sums of the function y = x2
Darboux lower sums of the function y = x2
Integral example with irregular partitions (largest marked in red)
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THE LIMIT DEFINITION OF A DEFINITE INTEGRAL
The following problems involve the limit definition of the definite integral of a continuous function of one variable on a closed, bounded interval. Begin with a continuous function on the interval . Let
be an arbitrary (randomly selected) partition of the interval , which divides the interval into subintervals (subdivisions). Let
be the sampling numbers (or sampling points) selected from the subintervals. That is,
is in ,
is in ,
is in , ... ,
is in ,
is in ,
is in .
Define the mesh of the partition to be the length of the largest subinterval. That is, let
for and define
The definite integral of on the interval is most generally defined to be
For convenience of computation, a special case of the above definition uses subintervals of equal length and sampling points chosen to be the right-hand endpoints of the subintervals. Thus, each subinterval has length
for and the right-hand endpoint formula is
for . The definite integral of on the interval can now be alternatively defined by
We will need the following well-known summation rules.
- (n times) , where is a constant
- , where is a constant
Most of the following problems are average. A few are somewhat challenging. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by using the formulas given above in exactly the form that they are given. Solutions to the first eight problems will use equal-sized subintervals and right-hand endpoints as sampling points as shown in equations (*) and (**) above.
Click HERE to return to the original list of various types of calculus problems.
Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :
Duane Kouba 2000-06-08