The branch of mathematics in which the notion of an
integral,
its properties and methods of calculation are
studied. Integral calculus is intimately related to
differential calculus,
and together with it constitutes the foundation of mathematical analysis. The origin
of integral calculus goes back to the early period of development of
mathematics and it is related to the method of
exhaustion developed by the mathematicians of Ancient Greece (cf.
Exhaustion, method of).
This method arose in the solution of problems on calculating areas of
plane figures and surfaces, volumes of solid bodies,
and in the solution of certain problems in statistics and
hydrodynamics. It is based on the approximation of the objects
under consideration by stepped figures or bodies, composed of
simplest planar figures or special bodies (rectangles, parallelopipeds, cylinders, etc.).
In this sense, the method of exhaustion can be regarded as
an early method of integration. The greatest development of the method of
exhaustion in the early period was obtained in the works of
Eudoxus
(4th century B.C.)
and especially
Archimedes
(3rd century B.C.).
Its
subsequent application and perfection is associated with the names
of several scholars of the
15th–17th centuries.
The fundamental concepts and theory of integral and differential
calculus, primarily the relationship between differentiation and integration, as well
as their application to the solution of applied problems, were developed in the works
of
P. de Fermat,
I. Newton
and
G. Leibniz
at the end of
the
17th century.
Their investigations were the beginning of an intensive development
of mathematical analysis. The works of
L. Euler,
Jacob
and
Johann Bernoulli
and
J.L. Lagrange
played an essential role in its creation in the
18th century.
In
the
19th century,
in connection with the appearance of the notion of
a limit, integral calculus achieved a logically complete form (in the works of
A.L. Cauchy,
B. Riemann
and others). The development of the theory and methods
of integral calculus took place at the end of
19th century
and in the
20th century
simultaneously with research into measure theory (cf.
Measure),
which plays an essential role in integral calculus.
By means of integral calculus it became possible to solve by a unified
method many theoretical and applied problems, both new ones which earlier had
not been amenable to solution, and old ones that had previously required
special artificial techniques. The basic notions of integral calculus are two
closely related notions of the integral, namely the indefinite and the definite integral.
The
indefinite integral
of a given real-valued function on an interval on the real axis is
defined as the collection of all its primitives on that interval,
that is, functions whose derivatives are the given
function. The indefinite integral of a function
is denoted by
.
If
is some primitive of
,
then any other primitive of it has the form
,
where
is an arbitrary constant; one therefore writes
The operation of finding an indefinite integral is called
integration.
Integration is the operation inverse to that of differentiation:
The operation of integration is linear: If on some interval the indefinite integrals
exist, then for any real numbers
and
,
the following integral exists on this interval:
and equals
For indefinite integrals, the formula of
integration by parts
holds: If two functions
and
are differentiable on some interval and if the integral
exists, then so does the integral
,
and the following formula holds:
The
formula for change of variables
holds: If for two functions
and
defined on certain intervals, the composite function
makes sense and the function
is differentiable, then the integral
exists and equals (see
Integration by substitution)
A function that is continuous on some bounded interval has a primitive on
it and hence an indefinite integral exists for it. The problem of
actually finding the indefinite integral of a specified function is complicated by the
fact that the indefinite integral of an elementary function is not an elementary
function, in general. Many classes of functions are known for which
it proves possible to express their indefinite integrals in terms of
elementary functions. The simplest examples of these are integrals that are obtained
from a table of derivatives of the basic elementary functions (see
Differential calculus):
1)
,
;
2)
;
3)
,
,
;
in particular,
;
4)
;
5)
;
6)
;
7)
;
8)
;
9)
;
10)
;
11)
;
12)
;
13)
;
14)
,
;
15)
(when
is under the square root, it is assumed that
).
If the denominator of the integrand vanishes at some point, then
these formulas are valid only for those intervals inside which the denominator
does not vanish (see formulas 1, 2, 6, 7, 11, 13, 15).
The indefinite integral of a rational function over any interval on which
the denominator does not vanish is a composition of rational functions,
arctangents and natural logarithms. Finding the algebraic part of the indefinite
integral of a rational function can be achieved by the
Ostrogradski method.
Integrals of the following types can be reduced by means
of substitution and integration by parts to integration of rational functions:
where
are rational numbers; integrals of the form
(see
Euler substitutions);
certain cases of integrals of differential binomials (cf.
Differential binomial;
Chebyshev theorem on the integration of binomial differentials);
integrals of the form
(where
are rational functions); the integrals
and many others. In contrast, for example, the integrals
cannot be expressed in terms of elementary functions.
The
definite integral
of a function
defined on an interval
is the limit of integral sums of a specific type (see
Cauchy integral;
Riemann integral;
Lebesgue integral;
Kolmogorov integral;
Stieltjes integral;
etc.). If this limit exists,
is said to be Cauchy, Riemann, Lebesgue, etc.
integrable.
The geometrical meaning of the integral is tied up
with the notion of area: If the function
is continuous on the interval
,
then the value of the integral
is equal to the area of the curvilinear trapezium formed by the graph of
the function, that is, the set whose boundary consists of the graph of
,
the segment
and the two segments on the lines
and
making the figure closed, which may degenerate to points (cf.
Fig.).
Figure: i051360a
The calculation of many quantities encountered in practice reduces to the
problem of calculating the limit of integral sums; in other words, finding
a definite integral; for example, areas of figures and surfaces, volumes of
bodies, work done by force, the coordinates of the centre of gravity,
the values of the moments of inertia of various bodies, etc.
The definite integral is linear: If two functions
and
are integrable on an interval
,
then for any real numbers
and
the function
is also integrable on this interval and
Integration of a function over an interval has
the property of monotonicity: If the function
is integrable on the interval
and if
,
then
is integrable on
as well. The integral is also additive with respect to
the intervals over which the integration is carried out: If
and the function
is integrable on the intervals
and
,
then it is integrable on
,
and
If
and
are Riemann integrable, then their product is also Riemann integrable. If
on
,
then
If
is integrable on
,
then the absolute value
is also integrable on
if
,
and
By definition one sets
A mean-value theorem holds for integrals. For example, if
and
are Riemann integrable on an interval
,
if
,
,
and if
does not change sign on
,
that is, it is either non-negative or non-positive
throughout this interval, then there exists a number
for which
Under the additional hypothesis that
is continuous on
,
there exists in
a point
for which
In particular, if
,
then
Integrals with a variable upper limit.
If a function
is Riemann integrable on an interval
,
then the function
defined by
is continuous on this interval. If, in addition,
is continuous at a point
,
then
is differentiable at this point and
.
In other words, at the points of continuity of a function the following formula holds:
Consequently, this formula holds for every Riemann-integrable function on an interval
,
except perhaps at a set of points having Lebesgue measure
zero, since if a function is Riemann integrable on some interval, then
its set of points of discontinuity has measure zero. Thus, if the function
is continuous on
,
then the function
defined by
is a primitive of
on this interval. This theorem shows that the operation of
differentiation is inverse to that of taking the definite integral with a
variable upper limit, and in this way a
relationship is established between definite and indefinite integrals:
The geometric meaning of this relationship is that the problem of finding the
tangent to a curve and the calculation of the area
of plane figures are inverse operations in the above sense.
The following
Newton–Leibniz formula
holds for any primitive
of an integrable function
on an interval
:
It shows that the definite integral of a continuous function over some interval
is equal to the difference of the values at the end points of this
interval of any primitive of it. This formula is sometimes taken as
the definition of the definite integral. Then it is proved that the integral
introduced in this way is equal to the limit of the corresponding integral sums.
For definite integrals, the formulas for change of variables and
integration by parts hold. Suppose, for example, that the function
is continuous on the interval
and that
is continuous together with its derivative
on the interval
,
where
is mapped by
into
:
for
,
so that the composite
is meaningful in
.
Then, for
,
the following formulas for
change of variables
holds:
The
formula for integration by parts
is:
where the functions
and
have Riemann-integrable derivatives on
.
The Newton–Leibniz formula reduces the calculation of an indefinite integral to
finding the values of its primitive. Since the problem of finding
a primitive is intrinsically a difficult one, other methods of finding definite
integrals are of great importance, among which one
should mention the method of residues (cf.
Residue of an analytic function;
Complex integration, method of)
and the method of differentiation or integration with respect to the parameter of a
parameter-dependent integral.
Numerical methods for the approximate computation of integrals have also been developed.
Generalizing the notion of an integral to the case of unbounded functions and
to the case of an unbounded interval leads to the notion of the
improper integral,
which is defined by yet one more limit transition.
The notions of the indefinite and the definite integral carry
over to complex-valued functions. The representation of any holomorphic function
of a complex variable in the form of a
Cauchy integral
over a contour played an important role in
the development of the theory of analytic functions.
The generalization of the notion of the definite integral of a function of a
single variable to the case of a function of several variables leads to the notion of a
multiple integral.
For unbounded sets and unbounded functions of several variables, one is led
to the notion of the improper integral, as in the one-dimensional case.
The extension of the practical applications of integral calculus
necessitated the introduction of the notions of the
curvilinear integral,
i.e. the integral along a curve, the
surface integral,
i.e. the integral over a surface, and more
generally, the integral over a manifold, which are reducible
in some sense to a definite integral (the curvilinear integral reduces to an
integral over an interval, the surface integral to an
integral over a (plane) region, the integral over an
-dimensional
manifold to an integral over an
-dimensional
region). Integrals over manifolds, in particular curvilinear and surface
integrals, play an important role in the integral calculus of
functions of several variables; by this means a relationship is established
between integration over a region and integration over its boundary or, in
the general case, over a manifold and its
boundary. This relationship is established by the
Stokes formula
(see also
Ostrogradski formula;
Green formulas),
which is a generalization of the Newton–Leibniz formula to the multi-dimensional case.
Multiple, curvilinear and surface integrals find direct application in
mathematical physics, particularly in field theory. Multiple integrals and concepts related
to them are widely used in the solution of
specific applied problems. The theory of cubature formulas (cf.
Cubature formula)
has been developed for the numerical calculation of multiple integrals.
The theory and methods of integral calculus of real- or complex-valued
functions of a finite number of real or complex variables carry over
to more general objects. For example, the theory of integration of functions
whose values lie in a normed linear space, functions defined on
topological groups, generalized functions, and functions of an infinite number
of variables (integrals over trajectories). Finally, a new direction in
integral calculus is related to the
emergence and development of constructive mathematics.
Integral calculus is applied in many branches of mathematics (in the
theory of differential and integral equations, in probability theory and
mathematical statistics, in the theory of optimal processes, etc.),
and in applications of it. For references see also
[1]–
to
Differential calculus.