Integral
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A definite integral of a function can be represented as the signed area of the region bounded by its graph.
Integration is an important concept in
mathematics and, together with its inverse,
differentiation, is one of the two main operations in
calculus. Given a
function f of a
real variable x and an
interval [a, b] of the
real line, the
definite integral
is defined informally to be the
area of the region in the
xy-plane bounded by the
graph of
f, the
x-axis, and the vertical lines
x = a and
x = b, such that area above the
x-axis adds to the total, and that below the
x-axis subtracts from the total.
The term
integral may also refer to the 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 termed
definite integrals.
The principles of integration were formulated independently by
Isaac Newton and
Gottfried Leibniz in the late 17th century. Through the
fundamental theorem of calculus, which they independently developed, integration is connected with differentiation: 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
Integrals and derivatives became the basic tools of calculus, with numerous applications in science and
engineering. The founders of the calculus thought of the integral as an infinite sum of rectangles of
infinitesimal width. A rigorous mathematical definition of the integral was given by
Bernhard Riemann. It is based on a limiting procedure which 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. Integrals of
differential forms play a fundamental role in modern
differential geometry. These generalizations of integrals first arose from the needs of
physics, and they play an important role in the formulation of many physical laws, notably those of
electrodynamics.
There are many modern concepts of integration, among these, the most
common is based on the abstract mathematical theory known as
Lebesgue integration, developed by
Henri Lebesgue.
History
Pre-calculus integration
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 shapes 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. 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
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 16th 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.
Formalizing integrals
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.
Historical notation
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
or
,
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 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).
Terminology and notation
The simplest case, the integral over
x of a real-valued function
f(
x), is written as
The integral sign ∫ represents integration. The
dx indicates that we are integrating over
x;
dx is called the
variable of integration. In correct mathematical typography, the
dx is separated from the integrand by a space (as shown). Some authors use an upright
d (that is, d
x instead of
dx). Inside the ∫...
dx is the expression to be integrated, called the
integrand. In this case the integrand is the function
f(
x). Because there is no domain specified, the integral is called an
indefinite integral.
When integrating over a specified domain, we speak of a
definite integral. Integrating over a domain
D is written as
or 
if the domain is an interval [a, b] of x;
The domain
D or the interval [
a,
b] is called the
domain of integration.
If a function has an integral, it is said to be
integrable. 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 (such as a
sample space in probability theory).
In the
modern Arabic mathematical notation,
which aims at pre-university levels of education in the Arab world and
is written from right to left, a reflected integral symbol
is used (
W3C 2006).
The variable of integration
dx has different interpretations depending on the theory being used. It can be seen as strictly a notation indicating that
x is a
dummy variable of integration; if the integral is seen as a
Riemann sum,
dx is a reflection of the weights or widths
d of the intervals of
x; in
Lebesgue integration and its extensions,
dx is a
measure; in
non-standard analysis, it is an
infinitesimal; or it can be seen as an independent mathematical quantity, a
differential form. More complicated cases may vary the notation slightly. In Leibniz's notation,
dx is interpreted an infinitesimal change in
x, but his interpretation
lacks rigour
in the end. Nonetheless Leibniz's notation is the most common one
today; and as few people are in need of full rigour, even his
interpretation is still used in many settings.
Introduction
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.
Approximations to integral of √
x from 0 to 1, with
■ 5 right samples (above) and
■ 12 left samples (below)
To start off, consider the curve
y = f(x) between
x = 0 and
x = 1 with
f(x) = √x. We ask:
- What is the area under the function f, in the interval from 0 to 1?
and call this (yet unknown) area the
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. As it is, the true value of the integral must be
somewhat less. Decreasing the width of the approximation rectangles
shall give 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
Notice that 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 never be exact:
replacing the 5 subintervals by twelve as depicted, we will get an
approximate value for the area of 0.6203, which is too small. 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.
As for the
actual calculation of integrals, the
fundamental theorem of calculus, due to Newton and Leibniz, is the fundamental link between the operations of
differentiating and integrating. Applied to the square root curve,
f(
x) =
x1/2, it says to look at the
antiderivative F(x) = (2/3)x3/2, and simply take
F(1) −
F(0), where 0 and 1 are the boundaries of the
interval [0,1]. So the
exact value of the area under the curve is computed formally as
(This is a case of a general rule, that for
f(x) = xq, with
q ≠ −1, the related function, the so-called
antiderivative is
F(x) = xq + 1/(q + 1).)
The notation
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. The multiplication sign is usually omitted.
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.
Differential geometry, with its "calculus on
manifolds", gives the familiar notation yet another interpretation. Now
f(
x) and
dx become a
differential form,
ω = f(x) dx, a new
differential operator d, known as the
exterior derivative is introduced, and the fundamental theorem becomes the more general
Stokes' theorem,
from which
Green's theorem, the
divergence theorem, and the
fundamental theorem of calculus follow.
More recently, infinitesimals have reappeared with rigor, through modern innovations such as
non-standard analysis. Not only do these methods vindicate the intuitions of the pioneers; they also lead to new mathematics.
Although there are differences between these conceptions of integral,
there is considerable overlap. Thus, the area of the surface of the
oval swimming pool can be handled as a geometric ellipse, a sum of
infinitesimals, a Riemann integral, a Lebesgue integral, or as a
manifold with a differential form. The calculated result will be the
same for all.
Formal definitions
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.
Riemann integral
Integral approached as Riemann sum based on tagged partition, with
irregular sampling positions and widths (max in red). True value is
3.76; estimate is 3.648.
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
Riemann sums converging as intervals halve, whether sampled at
■ right,
■ minimum,
■ maximum, or
■ left.
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.
Lebesgue integral
Riemann–Darboux's integration (blue) and Lebesgue integration (red).
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:
I have to pay a certain sum, which I have collected
in my pocket. I take the bills and coins out of my pocket and give them
to the creditor in the order I find them until I have reached the total
sum. This is the Riemann integral. But I can proceed differently. After
I have taken all the money out of my pocket I order the bills and coins
according to identical values and then I pay the several heaps one
after the other to the creditor. This is my integral.
- Source: (Siegmund-Schultze 2008)
As
Folland (1984, p. 56) puts it, "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 thus begins with a
measure, μ. In the simplest case, the
Lebesgue measure μ(
A) of an interval
A = [a,b] is its width,
b −
a,
so that the Lebesgue integral agrees with the (proper) Riemann integral
when both exist. In more complicated cases, the sets being measured can
be highly fragmented, with no continuity and no resemblance to
intervals.
Using the "partitioning the range of
f" philosophy, the integral of a non-negative function
f : R → R should be the sum over
t of the areas between a thin horizontal strip between
y = t and
y = t + dt. This area is just
μ{ x : f(x) > t} dt. Let
f∗(t) = μ{ x : f(x) > t}. The Lebesgue integral of
f is then defined by (
Lieb & Loss 2001)
where the integral on the right is an ordinary improper Riemann integral (note that
f∗
is a strictly decreasing positive function, and therefore has a
well-defined improper Riemann integral). For a suitable class of
functions (the
measurable functions) this defines the Lebesgue integral.
A general measurable function
f is Lebesgue integrable if the area between the graph of
f and the
x-axis is finite:
In that case, the integral is, as in the Riemannian case, the difference between the area above the
x-axis and the area below the
x-axis:
where
Other integrals
Although the Riemann and Lebesgue integrals are the most widely used
definitions of the integral, a number of others exist, including:
- The Darboux integral which is equivalent to a Riemann integral,
meaning that a function is Darboux-integrable if and only if it is
Riemann-integrable, and the values of the two integrals, if they exist,
are equal. Darboux integrals have the advantage of being simpler to
define than Riemann integrals.
- The Riemann–Stieltjes integral, an extension of the Riemann integral.
- The Lebesgue-Stieltjes integral, further developed by Johann Radon, which generalizes the Riemann–Stieltjes and Lebesgue integrals.
- The Daniell integral, which subsumes the Lebesgue integral and Lebesgue-Stieltjes integral without the dependence on measures.
- The Haar integral, used for integration on locally compact topological groups, introduced by Alfréd Haar in 1933.
- The Henstock–Kurzweil integral, variously defined by Arnaud Denjoy, Oskar Perron, and (most elegantly, as the gauge integral) Jaroslav Kurzweil, and developed by Ralph Henstock.
- The Itō integral and Stratonovich integral, which define integration with respect to semimartingales such as Brownian motion.
- The Young integral, which is a kind of Riemann–Stieltjes integral with respect to certain functions of unbounded variation.
- The rough path
integral defined for functions equipped with some additional "rough
path" structure, generalizing stochastic integration against both semimartingales and processes such as the fractional Brownian motion.
Properties
Linearity
- The collection of Riemann integrable functions on a closed interval [a, b] forms a vector space under the operations of pointwise addition and multiplication by a scalar, and the operation of integration
-
- is a linear functional on this vector space. Thus, firstly, the collection of integrable functions is closed under taking linear combinations; and, secondly, the integral of a linear combination is the linear combination of the integrals,
-
- Similarly, the set of real-valued Lebesgue integrable functions on a given measure space E with measure μ is closed under taking linear combinations and hence form a vector space, and the Lebesgue integral
-
- is a linear functional on this vector space, so that
-
-
- that is compatible with linear combinations. In this situation the
linearity holds for the subspace of functions whose integral is an
element of V (i.e. "finite"). The most important special cases arise when K is R, C, or a finite extension of the field Qp of p-adic numbers, and V is a finite-dimensional vector space over K, and when K=C and V is a complex Hilbert space.
Linearity, together with some natural continuity properties and
normalisation for a certain class of "simple" functions, may be used to
give an alternative definition of the integral. This is the approach of
Daniell for the case of real-valued functions on a set
X, generalized by
Nicolas Bourbaki to functions with values in a locally compact topological vector space. See (
Hildebrandt 1953) for an axiomatic characterisation of the integral.
Inequalities for integrals
A number of general inequalities hold for Riemann-integrable
functions defined on a
closed and
bounded interval [
a,
b] and can be generalized to other notions of integral (Lebesgue and Daniell).
- Upper and lower bounds. An integrable function f on [a, b], is necessarily bounded on that interval. Thus there are real numbers m and M so that m ≤ f (x) ≤ M for all x in [a, b]. Since the lower and upper sums of f over [a, b] are therefore bounded by, respectively, m(b − a) and M(b − a), it follows that
-
- Inequalities between functions. If f(x) ≤ g(x) for each x in [a, b] then each of the upper and lower sums of f is bounded above by the upper and lower sums, respectively, of g. Thus
-
- This is a generalization of the above inequalities, as M(b − a) is the integral of the constant function with value M over [a, b].
- In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if f(x) < g(x) for each x in [a, b], then
- Subintervals. If [c, d] is a subinterval of [a, b] and f(x) is non-negative for all x, then
-
-
- If f is Riemann-integrable on [a, b] then the same is true for |f|, and
- Moreover, if f and g are both Riemann-integrable then f 2, g 2, and fg are also Riemann-integrable, and
- This inequality, known as the Cauchy–Schwarz inequality, plays a prominent role in Hilbert space theory, where the left hand side is interpreted as the inner product of two square-integrable functions f and g on the interval [a, b].
- Hölder's inequality. Suppose that p and q are two real numbers, 1 ≤ p, q ≤ ∞ with 1/p + 1/q = 1, and f and g are two Riemann-integrable functions. Then the functions |f|p and |g|q are also integrable and the following Hölder's inequality holds:
- For p = q = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality.
- Minkowski inequality. Suppose that p ≥ 1 is a real number and f and g are Riemann-integrable functions. Then |f|p, |g|p and |f + g|p are also Riemann integrable and the following Minkowski inequality holds:
- An analogue of this inequality for Lebesgue integral is used in construction of Lp spaces.
Conventions
In this section
f is a
real-valued Riemann-integrable
function. The integral
over an interval [
a,
b] is defined if
a <
b. This means that the upper and lower sums of the function
f are evaluated on a partition
a = x0 ≤ x1 ≤ . . . ≤ xn = b whose values
xi are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating
f within intervals [
x i ,
x i +1] where an interval with a higher index 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
[a, 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
[a, 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 [a, 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 of differential forms 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:
These conventions correspond to interpreting the integrand as a differential form, integrated over a
chain. In
measure theory, by contrast, one interprets the integrand as a function
f with respect to a measure
and integrates over a subset
A, without any notion of orientation; one writes
![\textstyle{\int_A f\,d\mu = \int_{[a,b]} f\,d\mu}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vdPst2Piq8qwWbMaqDKU1s9en8YVf6TaGdfXTaHIZq_qvjw4ItnX0Y3Lyf2mfD-NIkutqVYmhkL7SEzT6gYe6b7YPy6pSIf1RGEJCFAu2bd0DCKrF-zoncx6FhUm7WXgXb14LJk36KqWWpLSvuCg=s0-d)
to indicate integration over a subset
A. This is a minor distinction in one dimension, but becomes subtler on higher dimensional manifolds; see
Differential form: Relation with measures for details.
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 continuous real-valued function defined on a closed interval [a, b]. Let F be the function defined, for all x in [a, b], by
Then,
F is continuous on [
a,
b], differentiable on the open interval
(a, b), and
for all
x in (
a,
b).
- Second fundamental theorem of calculus. Let f be a real-valued function defined on a closed interval [a, b] that admits an antiderivative g on [a, b]. That is, f and g are functions such that for all x in [a, b],
If
f is integrable on
[a, b] then
Extensions
Improper integrals
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.
That is, 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, for example, 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, 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,
. This has a finite limit as
t
goes to infinity, namely π/2. Similarly, the integral from 1/3 to 1
allows a Riemann sum as well, coincidentally again producing π/6.
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 π/2. Combining the limits of the two fragments, the result of this improper integral is
This process does not guarantee success; a limit may fail to exist,
or may be unbounded. For example, over the bounded interval 0 to 1 the
integral of 1/
x 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
![\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}](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_uaZUoKfrzeXSGCDlQL1dwnHgPwxgqwrrHO_ixaHEOo4TW48Ax-Jdimg-PsLaiTK97XaFGFX5wVAB38UXuqQCrJ8RxWyV4IsB3mwWU1Fm9MvsuLeQJHgLtVI9HGoTvlquUh5176KW2F4FG9Nw1i=s0-d)
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
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:
-
- 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 3 ≤ x ≤ 7, 4 ≤ y ≤ 10, our above double integral now reads
-
![\int_4^{10}\left[ \int_3^7 \ 5 \ dx\right] dy](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_si4zHgDohMPK_vmkqINvQX6LBBv8ttH4tjqxIhkVw7XuA1blPOPyRRo6mUKcZM53m-6u2IN_Uo9F6F9Rfvd-IHk2gWaFNWPD8zo3s8f4qSTZvFm7J4PZU285vTgv4Uoe98wT2FvXTtlzyeRDUT=s0-d)
- 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.
-
- of the constant function 1 calculated on the cuboid itself.
Line integrals
Main article:
Line integral
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,
F, multiplied by displacement,
s, may be expressed (in terms of vector quantities) as:
For an object moving along a path in a
vector field 
such as an
electric field or
gravitational field, the total work done by the field on the object is obtained by summing up the differential work done in moving from
to
. This gives the line integral
Surface integrals
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
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 and associativity 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 two-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
Stokes' theorem and the
divergence theorem. In this way we can see that differential forms provide a powerful unifying view of integration.
Summations
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.
Methods
Computing integrals
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]. Then, find an antiderivative of
f; that is, a function
F such that
F' =
f on the interval. 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. Techniques include:
Alternate 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.
Symbolic algorithms
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. For instance, 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; in other words, none of the three given functions is integrable in
elementary functions, which are the functions which may be built from rational functions,
roots of a polynomial, logarithm, and exponential functions. 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.
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, the
Incomplete Gamma function and so on — see
Symbolic integration
for more details). Extending the Risch's algorithm to include such
functions is possible but challenging and has been an active research
subject.
More recently a new approach has emerged, using
D-finite function, which are the solutions of
linear differential equations with polynomial coefficients. Most of the elementary and special functions are
D-finite and the integral of a
D-finite function is also a
D-finite function. This provide an algorithm to express the antiderivative of a
D-finite function as the solution of a differential equation.
This theory allows also to compute a definite integrals of a
D-function as the sum of a series given by the first coefficients and an algorithm to compute any coefficient.
[1]
Numerical quadrature
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 2008;
Kahaner, Moler & Nash 1989;
Stoer & Bulirsch 2002). Consider, for example, 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, 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, 2
18 pieces are
required, a great computational expense for such 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. Furthermore, only 2
10 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. 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 (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. In this example, 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 2
n−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 |
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 formulae.
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.
Simpson's rule, named for
Thomas Simpson (1710–1761), uses a parabolic curve to approximate integrals. In many cases, it is more accurate than the
trapezoidal rule and others. The rule states that
![\int_a^b f(x) \, dx \approx \frac{b-a}{6}\left[f(a) + 4f\left(\frac{a+b}{2}\right)+f(b)\right],](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_veyRqORUFa6Hx-7IytRaFUMcfNS_BymMjkELsyTemueVUlj12LX7Logtvn6o_ekGj_AWTCaxp-VlIE-rTRGafIZeyy7_0KYIkILL7S9agSVs-U23u-0h55TORKDLwlkB91OBYaElYmU0gNYQb28Q=s0-d)
with an error of
The computation of higher-dimensional integrals (for example, volume calculations) makes important use of such alternatives as
Monte Carlo integration.
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.
See also
Notes
References
- Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), Wiley, ISBN 978-0-471-00005-1
- Bourbaki, Nicolas (2004), Integration I, Springer Verlag, ISBN 3-540-41129-1. In particular chapters III and IV.
- Burton, David M. (2005), The History of Mathematics: An Introduction (6th ed.), McGraw-Hill, p. 359, ISBN 978-0-07-305189-5
- Cajori, Florian (1929), A History Of Mathematical Notations Volume II, Open Court Publishing, pp. 247–252, ISBN 978-0-486-67766-8
- Dahlquist, Germund; Björck, Åke (2008), "Chapter 5: Numerical Integration", Numerical Methods in Scientific Computing, Volume I, Philadelphia: SIAM
- Folland, Gerald B. (1984), Real Analysis: Modern Techniques and Their Applications (1st ed.), 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
Available in translation as Fourier, Joseph (1878), The analytical theory of heat, Freeman, Alexander (trans.), Cambridge University Press, pp. 200–201
- Heath, T. L., ed. (2002), The Works of Archimedes, Dover, ISBN 978-0-486-42084-4
(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
- Kahaner, David; Moler, Cleve; Nash, Stephen (1989), "Chapter 5: Numerical Quadrature", Numerical Methods and Software, Prentice Hall, ISBN 978-0-13-627258-8
- Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2
- Leibniz, Gottfried Wilhelm (1899), Gerhardt, Karl Immanuel, ed., Der Briefwechsel von Gottfried Wilhelm Leibniz mit Mathematikern. Erster Band, Berlin: Mayer & Müller
- Lieb, Elliott; Loss, Michael (2001), Analysis (2 ed.), AMS Chelsea, ISBN 978-0821827833
- Miller, Jeff, Earliest Uses of Symbols of Calculus, retrieved 2009-11-22
- O’Connor, J. J.; Robertson, E. F. (1996), A history of the calculus, retrieved 2007-07-09
- Rudin, Walter (1987), "Chapter 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. Young. With two additional notes by Stefan Banach. Second revised ed.), New York: Dover
- Shea, Marilyn (May 2007), Biography of Zu Chongzhi, University of Maine, retrieved 9 January 2009
- Siegmund-Schultze, Reinhard (2008), "Henri Lebesgue", in Timothy Gowers, June Barrow-Green, Imre Leader, Princeton Companion to Mathematics, Princeton University Press.
- Stoer, Josef; Bulirsch, Roland (2002), "Chapter 3: Topics in Integration", Introduction to Numerical Analysis (3rd ed.), Springer, ISBN 978-0-387-95452-3.
- W3C (2006), Arabic mathematical notation
External links
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, Benjamin, Calculus, 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
| [hide]
Integrals
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| Methods |
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| Improper Integrals |
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| Stochastic integrals |
|
Ok dech… Kali ini kita akan membahas tentang Penjumlahan Vektor, yang merupakan bahasan lanjutan dari Pengukuran Besaran Fisika.
097. Padanya terdapat tanda-tanda yang nyata, (di antaranya) maqam Ibrahim; barangsiapa memasukinya (Baitullah itu) menjadi amanlah dia; mengerjakan haji adalah kewajiban manusia terhadap Allah, yaitu (bagi) orang yang sanggup mengadakan perjalanan ke Baitullah; Barangsiapa mengingkari (kewajiban haji), maka sesungguhnya Allah Maha Kaya (tidak memerlukan sesuatu) dari semesta alam.(Ali Imron 3:097) 
is used (W3C 2006).
![\int_{-1}^{1} \frac{dx}{\sqrt[3]{x^2}} = 6](https://lh3.googleusercontent.com/blogger_img_proxy/AEn0k_vtDCGTEVqhDDVryIGLhbiyDdru-cWsQxY5Zy9cFxdtpIAcwrbOkihU1tZEHY_JNlidiI6RLyXoofgFe2vqgyX4CJJWJAdesbasmeBnt6_cmXeZA65paTwUGL94lDp0q5FmUJ_fUZKzmlq1tVu3lQ=s0-d)
