Integral calculus

The calculus is in addition to the differential calculus of the most important branch of mathematical discipline of analysis. It arose from the problem of area and volume calculation. The integral is a generic term for the indefinite and the definite integral. The calculation of integrals is called integration.

The definite integral of a function assigns these to a numerical value. By forming the definite integral of a real function in a variable, so the result can be in two-dimensional coordinate system as the area of ​​the surface that lies between the graph of the function, the axis and the limiting parallels to the axis point. These include patches below the negative axis. One speaks of the oriented area. This convention is chosen so that the definite integral gives a linear map, which is a central feature of the integral term for both theoretical considerations as well as concrete computations.

The indefinite integral of a function assigns these to a set of functions whose elements are called primitives. These are characterized by the fact that their first derivatives with the function that has been integrated match. The fundamental theorem of differential and integral calculus states that (certain) integrals can be calculated from ordinary functions.

In contrast to the differentiation there is no easy and not all cases a concealing algorithm for the integration of elementary functions also. Integration requires trained rates, use of special transformations ( integration by substitution, integration by parts ), looking up in an integral table or using special computer software. Often, the integration is carried out only approximately by means of so-called numerical quadrature.

In the art is used for the approximate determination of the so-called surface planimeter, in which the summation of the area elements is carried out continuously. The numerical value of the area thus determined can be read on a counter, which is provided to increase the accuracy of reading with a vernier. Chemists were wont formerly integrals of arbitrary surfaces using an analytical balance or microbalance to determine: The area was carefully excised and weighed, as well as a exactly 10 cm × 10 cm piece of the same paper; a three calculation led to the result.

• 3.1 Properties of common functions
• 3.2 Indefinite Integral
• 3.3 Determination of primitives 3.3.1 Integration by parts
• 3.3.2 Integration by substitution
• 3.3.3 Forming by partial fractions
• 3.3.4 Special procedures

• 4.1 Mean values ​​of continuous functions
• 4.2 Example for the integral term in physics

• 5.1 Cauchy's integral
• 5.2 Riemann integral
• 5.3 Stieltjes integral
• 5.4 Lebesgue integral

• 7.1 Numerical Methods
• 7.2 Exact method
• 7.3 Special integrals

• 8.1 path integrals 8.1.1 Real path integrals and length of a curve
• 8.1.2 Real path integrals: With scalar
• 8.1.3 Complex path integrals

• 8.2.1 Example: Calculation of spatial content
• 8.2.2 surface integrals 8.2.2.1 Integration over a chart domain
• 8.2.2.2 integration over a submanifold
• 8.2.2.3 The Gaussian integral theorem and the theorem of Stokes

• 9.1 Measure Theory
• 9.2 Hair cal measure
• 9.3 Integration on manifolds

History

Area calculations are studied since ancient times. In the 5th century BC, Eudoxus of Cnidus developed from an idea by Antiphon the method of exhaustion, which was to estimate ratios of surface areas contained by or of overlapping polygons. He was able to determine by this method both surface areas and volumes of some simple body. Archimedes ( 287-212 BC ) improved this approach, and so he was able to determine the exact contents of a surface of a parabola and a secant limited area without recourse to the limit term that did not exist at that time; This result can easily be transformed into the now well-known integral of a quadratic function. He also estimated the ratio of circumference to diameter, as a value between up and down.

This method was also used in medieval times. In the 17th century Bonaventura Francesco Cavalieri presented on the principle of Cavalieri, that two bodies have the same volume, if all parallel plane sections have the same area. Johannes Kepler used in his work Astronomia Nova (1609 ) in calculating the orbit of Mars methods that would be now known as numerical integration. He tried from 1612 to calculate the volume of wine barrels. In 1615 he published the Stereometria Doliorum vinariorum ( " stereometry the wine barrels " ), later also known as Kepler Fassregel known.

End of the 17th century did Isaac Newton and Gottfried Wilhelm Leibniz to develop independently calculi to differential calculus and thus to discover the Fundamental Theorem of Calculus ( history of discovery and priority dispute see the article calculus, the integral sign and its history, see integral sign ). Her work has allowed the abstraction of purely geometric notion and are therefore considered as the beginning of analysis. They became known primarily through the book of the nobles Guillaume François Antoine, Marquis de L' Hospital, who took private lessons from Johann Bernoulli and his research on analysis published so. The term integral goes back to Johann Bernoulli.

In the 19th century, the entire analysis was placed on a more solid foundation. Augustin- Louis Cauchy in 1823 developed the first time an integral concept that meets the current requirements of stringency. Later the terms of the Riemann integral and the Lebesgue integral emerged. Finally, the development of measure theory followed at the beginning of the 20th century.

Integral for compact intervals

"Compact " means closed and bounded, so it only functions on intervals of the form are considered. Open or unrestricted intervals are not allowed.

Motivation

Reduction of more complicated surface content on integrals

An object of the integral calculus is the computation of surface areas curvilinear areas of the layer. In most cases occurring in practice such surfaces are described by two continuous functions on a compact interval, the graph, the limit surface (left image).

The area of ​​the shaded area in the left image is equal to the difference between the shaded areas in the two right images. It is therefore sufficient to be limited to the simpler case of a surface which is bounded by:

• The graph of a function,
• Two vertical lines and
• And the axis.

Due to its immense importance of this area type given a special name:

Read as an integral of up to about (or by). Instead, a different variable, apart from and are selected, for example, which does not change the value of the integral.

Integrals of nonnegative functions

If one shifts the graph of a function in the direction of the axis by a piece, then come to the surface considered a rectangle to:

The integral changes by the surface area of this rectangle the width and height, in formulas

If one considers a continuous function whose values ​​are negative, then we can always find a so that all the values ​​in the interval (to greater than the amount of the minimum of his ) are positive. With the above consideration, we obtain

That is, the integral of the difference of the areas of the white area in the center and the surrounding rectangle. This difference, however, is negative, that is, to the above formula for any function to be correct, so you have to include negative areas below the axis. We therefore speak of an oriented or directed area.

If one or more zeros present in the to be tested interval, the integral is not the area of, but the sum of the (positive) surface areas of the surface portions above the axis and the (negative) surface areas of the sub- areas below the axis. Does one need in such an interval, the area between axis and graph of the function, the integral of the zeros must be divided.

The principle of Cavalieri and the additivity of the integral

Axiomatic access

It is not easy to grasp the concept of the area of ​​mathematical precision. Over time, various concepts have been developed for this. However, the details of which are not necessary for most applications, since they coincide, inter alia, on the category of continuous functions. The following are some properties of the integral are listed, which were motivated above and independent of the exact design standards apply for each integral. They also specify the integral of continuous functions clearly defined.

There are real numbers, and it was a vector space of functions, comprising the continuous functions. Functions are called " integrated ". Then, an integral is a figure

Written

Having the following characteristics:

• Linearity: For functions and,
• .

Designations

• The real numbers are called and integration limits. They can be located above and below or to the side of the integral character written by the integral sign:

• The function to be integrated is called the integrand.
• The variable is called variable of integration. If the integration variable, we also speak of integration over. The integration variable is replaceable, instead

• The ingredient " " is called differential, but in that context usually only symbolic significance. Consequently, no attempt here to define it. On differential one reads the integration variable.

Origin of the notation

The symbolic notation of integrals goes to the co-discoverer of the differential and integral calculus, Gottfried Wilhelm Leibniz, back. The integral sign ∫ is derived from the letter long s (s ) for Latin summa. The multiplicative -to-read notation indicates how the integral - the Riemann integral following - composed of strips of height and infinitesimal width.

Alternative notation in physics

In theoretical physics, for pragmatic reasons, often a slightly different notation for integrals used (especially in case of multiple integrals ). There is instead

Often

Written, sometimes both conventions are used in various places.

The second example has the disadvantage that the function to be integrated is not longer and bracketed. In addition, misunderstandings can occur, for example the Lebesgue integral. However, the alternative spelling also has some advantages:

• The term emphasizes that the integral is a linear operator that acts on everything to the right of him.
• Often appear in physics on integrals, in which the function to be integrated is several lines long, or it is integrated over several unknowns. Then you know how to spell at the beginning of the integral which variables are integrated at all and what limits. Furthermore, it is the allocation of variables to limits is easier.
• The commutativity of the products at the summands appearing in the Riemann approximation is emphasized.

Example:

Instead of

Simple consequences of the axioms

• If for all, then

• If we denote by the supremum of, so true

• Is for those without a fixed number, it shall

• Integrals of step functions: If a step function, that is, is a disjoint union of intervals of lengths, so that is on constant with value, then applies

Primitives and the fundamental theorem of differential and integral calculus

, In a sense the inverse of the integration differentiation. In order to clarify this, the notion of primitive is required: If a function is the name of a function, a primitive function of when the derivative of the same is:

The fundamental theorem of differential and integral calculus establishes the relationship between antiderivatives and integrals. He said: If a continuous function on an interval and is a primitive function of, the following applies

The right side is often abbreviated as

Written.

This relationship is the primary method for the explicit evaluation of integrals. The difficulty is usually in finding an antiderivative.

The mere existence is assured in theory: the integral function

Is an antiderivative of each.

Properties of primitive functions

You can add up to a root function is a constant and regains a primitive function: Is a primitive function to a function and is a constant, then

Two primitives of the same defined on an interval function differ by a constant: Are and antiderivatives of the same function, so is

Thus, the difference is a constant. Is the domain of no interval, so the difference between two primitives is only locally constant.

Indefinite Integral

A primitive function is also called the indefinite integral of - sometimes it is thus the set of all primitive functions meant. If a primitive function, then one writes frequently imprecise

To suggest that any primitive function has the form of a constant. The constant is called constant of integration.

Note that the notation

But also often used in formulas to indicate that equations for arbitrary, consistent selected limits apply; For example, with

Meant that

For arbitrary.

Determination of root functions

See article: Table of derivation and antiderivatives or indefinite integrals in the formulary Mathematics

In contrast to the derivative function, the explicit calculation of a primitive function with many functions is very difficult or impossible. Oftentimes, you hit integrals in Table works (eg an integral panel) after. For manual calculation of a primitive function is often the skillful application of the following standard techniques helpful.

Integration by parts

The partial integration is the inverse of the product rule of differential calculus. It reads:

This rule is advantageous when the function is simpler than the function to be integrated. However, this examination to evaluate the products and not the factors themselves.

Example:

Substituting

So is

And one obtains

Integration by substitution

The substitution rule is an important tool to calculate some difficult integrals, since it allows certain changes function to be integrated with simultaneous change of the integration limits. It is the counterpart to the chain rule of differential calculus.

Be with and an antiderivative of, so is a primitive function of, because it is

And with the substitution

Finally

Transformation by partial fractions

In fractional rational functions often performs a polynomial or a partial fraction expansion to a transformation of the function that allows to apply the rules of integration.

Specific methods

Often, it is possible to determine the primitive by utilizing the particular form of the integrand.

A further possibility is to start at a known integral and this transform by integration techniques until the desired integral is formed. example:

To determine, we integrate the following similar integral partial:

Rearranging follows

Applications

Mean values ​​of continuous functions

To calculate the mean value of a given continuous function at the interval, one uses the formula

As this definition matches for step functions with the usual averaging term, this generalization is useful.

The mean value theorem of calculus states that this average is actually accepted by a continuous function in the interval.

Example, for the integral term in physics

A physical phenomenon, in which the integral term can be explained, is the free fall of a body in the gravitational field of the earth. As is known, is the acceleration of free fall in Central Europe about 9.81 m / s ². The velocity of a body at the time, therefore, can be represented by the formula

Express.

But the path to be computed, the travels of the falling body within a certain time. The problem here is that the speed of the body increases with time. To solve the problem, it is assumed that the speed resulting from the time remains constant for a short period of time.

The increase of the distance within the short period of time is therefore

The entire route can therefore be as

Express. If you leave now seek the time difference to zero, one obtains

The integral can be analytically specify with

The general solution leads to the equation of motion of the falling body in a constant gravitational field:

Next can be calculated from this equation of motion by differentiating with respect to time the equation for the velocity:

And derived by again differentiating for the acceleration:

Other simple examples are:

• The power is the integral of power over time.
• The electric charge of a capacitor is the integral of the current flowing through over time.
• The integral of the product of the spectral irradiance ( Ee ( ν ) in W/m2Hz ) with the spectral luminosity curve of the eye provides the illuminance (E in lux = lumen/m2 ).
• The integral of the velocity ( longitudinal component) over the cross section of the tube provides the entire flow through the tube ( other multi-dimensional integrals see below).

Constructions

Cauchy's integral

A rule function is a function defined by step functions can be approximated uniformly. Due to the aforementioned compatibility of the integral with uniform Limites there is a control function, the uniform limit is a sequence of step functions, which define the integral as

Wherein the integral of step functions defined by the formula above.

The class of control functions comprises all continuous functions and all monotone functions, as well as all functions that can be subdivided into finitely many intervals, so that the restriction of a continuous or monotonic function on the closed interval. For many practical purposes, this integral construction is sufficient.

Riemann integral

An approach to the calculation of the integral is the approximation of the function to be integrated by a step function.

The surface is approximated by the sum of the areas of the individual rectangles with the individual " steps ". At any disassembly of the integration interval, one can choose an arbitrary function value within each sub-interval as height of the step to do so.

These are designated after the German mathematician Bernhard Riemann Riemann sums. If one chooses in each sub-interval of the decomposition just the supremum of the function as height of the rectangle, we obtain the upper sum, with the infimum of the lower sum.

The difference between the upper and lower sum can be explained by the product of the - estimate total variation and the maximum interval length in the cutting - also introduced by Riemann. Thus, the Riemannian subtotals converge to a definite integral value stated, if the width of the rectangles tends to zero and the total variation is finite. This limit can not be calculated explicitly for all functions or integral boundaries. The described method is referred to as a Riemann integration.

Functions bounded total variation are all continuous and piecewise continuous, and all monotone functions. Conversely, one can show that there can be for such functions only countably many points of discontinuity and that number thereof for each jump height is finite.

The Riemann integral can not be used in the integrand of infinite variation, for example, functions with oscillating singularities like or the indicator function of the rational numbers in the interval [ 0,1]. Therefore, advanced concepts of Henri Léon Lebesgue integral ( Lebesgue integral ), Jean Thomas Stieltjes were introduced ( Stieltjesintegral ) and Alfréd hair that reproduce the Riemann integral for continuous integrand.

Stieltjes integral

When Stieltjes integral to go from monotonous functions, or from those with finite variation, which are differences of two monotone functions, and sums defined for continuous functions, Riemann - Stieltjes'sche as

By Limes education in the usual way, we obtain the so-called Riemann - Stieltjes integral.

Such integrals are well defined if the function is not differentiable (otherwise applies ). A well-known counter-example is the so-called Heaviside function whose value is zero for negative numbers is unity for positive and for example for the point. One writes for and obtains the " generalized function ", the so-called Diracmaß than a defined only for the point measure.

Lebesgue integral

A more modern and - in many ways - better than the integral term of the Riemann integral yields the Lebesgue integral. It allows, for example, the integration over general measure spaces. This means that you can assign a measure quantities, which need not necessary coincide with their geometric length or their volume, such as probability measures in probability theory. The amount corresponding to the length or volume of intuitive notion is the Lebesgue measure. In general, the integral of this level is referred to as Lebesgue integral. One can prove that for any function that is over a compact interval Riemann integrable, the corresponding Lebesgue integral exists and the values ​​of both integrals coincide. Conversely, not all Lebesgue integrable functions Riemann integrable. The best known example is the Dirichlet function, so the function for rational numbers has the value one, but for irrational numbers to zero. In addition to the larger class of integrable functions, the Lebesgue integral over the Riemann integral is characterized mainly by the better convergence rates of ( set of the monotone convergence theorem of the dominated convergence ) and the superior characteristics of the normalized by the Lebesgue integral function rooms (eg completeness ).

In modern mathematics is meant by integral or integration theory often the Lebesgue integral term.

Improper integral

The Riemann integral is defined ( in the one-dimensional space ) only for compact, thus bounded and closed, intervals. A generalization to unbounded domains or functions with singularities offers the improper integral. In the Lebesgue theory of improper integrals can be considered, but this is not as productive as you can already integrate many functions with singularities or unbounded domain with the Lebesgue integral.

Method for calculating certain improper integrals and

Numerical Methods

Often it is difficult or not possible to explicitly specify a primitive function. However, it is in many cases also to calculate the definite integral approximation. This is called numerical quadrature or numerical integration. Many methods for the numerical quadrature build on an approximation of the function by a simple integrable functions on, for example, by polynomials. The trapezoidal rule or Simpson's formula and ( is the special case is known as Kepler Fassregel ) are examples of this is determined by the interpolation function determined and then integrated.

Long before the proliferation of computers method for automatic step size control were developed for the numerical integration. Today, the computer algebra provides the ability to complex integrals to be solved numerically in shorter time periods or more accurately. Whereby even in powerful systems still are difficulties in improper integrals. Frequently, special methods such as Gauss - Kronrod must be selected. An example of such a hard integral is:

Classic methods are, for example, the Euler's empirical formula, wherein the definite integral is approximated by a generally asymptotic series. Other methods are based on the theory of the calculus of finite differences, as an important example here is the Gregorysche integration formula to call.

Exact method

There are a number of methods by which definite and improper integrals can be accurately calculated in symbolic form.

If it is known to be a primitive function, can be the definite integral

Calculated by the main clause. The problem is that the operation of the indefinite integrating leads to an expansion given Funktionensklassen. For example, the integration within the class of rational functions not completed and lists the functions and. Also, the class of so-called elementary functions is not complete. So Joseph Liouville proved that the function has no elementary antiderivative. Leonhard Euler was one of the first who developed methods for accurate calculation of certain improper integrals and without fixing a primitive function. In the course of time a number of general and specific methods for the specific integration are created:

• Using the residue theorem
• Representation of a parameter -dependent integral by special functions
• Differentiation or integration of the integral with respect to a permutation of the parameters and boundary processes
• Use of a series expansion of the integrand with member -wise integration
• Attributed by integration by parts and substitution of the integral on yourself or another

By the end of the 20th century, numerous ( mostly multi-volume ) integral tables have been created with definite integrals. To illustrate the problem a few examples:

Particular integrals

There are a number of definite and improper integrals, which have some significance for mathematics and therefore bear its own name:

• Eulerian integrals of the first and second kind
• Gaussian error integral

• Fresnel integrals

• Raab Lagrangian integral

• Frullanische integrals

Multidimensional integration

Path integrals

Real path integrals and length of a curve

If one route, ie a continuous map, and a function, so the path integral along is defined as

Is, we obtain from the above formula, the length of the curve ( physically speaking ) as the integral of velocity over time:

Real path integrals: With scalar

In physics, path integrals of the form are commonly used: is a function, and it is the integral of

Considered.

Complex path integrals

In the theory of functions, ie the extension of calculus to functions of a complex variable, it is no longer sufficient to specify lower and upper limits of integration. Two points on the complex plane, unlike two points on the number line are connected to each other by many ways. Therefore, the definite integral in function theory is basically a path integral. For closed paths of the residue theorem, an important result of Cauchy applies: The integral of a meromorphic function along a closed path depends solely on the number of enclosed singularities. It is zero if there are no singularities in the region of integration.

Integration over multi-dimensional spaces

The integral term can be generalized to the case that the amount of carrier on which operates the integrand, not the number line, but the dimensional Euclidean space. Multidimensional integrals over a volume one can compute after the Fubini's theorem, by splitting them in any order integrals over the individual coordinates that have to be executed in sequence:

The integration limits of one-dimensional integrals, and must be determined from the boundary of the volume. Analogous to the improper integrals in the one-dimensional (see above) but you can also consider integrals over the entire, unlimited -dimensional space.

The generalization of the substitution rule in the multidimensional is the transform set. Be open and an injective, continuous differentiable map clauses apply to the Jacobian for all. Then

Example: Calculation of spatial content

As an example, the volume between the graph of the function is calculated over the unit square. For this purpose, one divides the integral over two integrals, one for the - and one for the coordinate:

Surface integrals

In particular, in many physical applications, the integration is not a volume, but on the surface of an area is of interest. Such surfaces are usually described by manifolds. These are described by so-called cards.

Integration via a card area

Be one - dimensional submanifold of the area and a card in, so an open subset in, for which there is a card that they diffeomorphic to an open subset of the maps. It should also be a parametrization of, ie a continuously differentiable mapping whose derivative has full rank, which is homeomorphic to mapping. Then, the integral of the function on the card area is defined as follows:

The Gram determinant is. The right-hand integral can be described with the above methods of multidimensional integration calculated. The equality follows essentially from the transform set.

Integration over a submanifold

Is a decomposition of the given one that is compatible with the maps of the submanifold can be easily integrated and summed separately over the map areas.

The Gaussian integral theorem and the theorem of Stokes

For special functions, the integrals over submanifolds are easier to calculate. In physics, especially important are two statements:

Firstly, the Gaussian integral theorem, according to which the volume integral of the divergence of a vector field is equal to the surface integral of the vector field ( the flow of the field through the surface ) is: Be compact with smooth boundary sections. The edge is oriented by an outward normal unit field. Further, let a continuous differentiable vector field on an open area of. Then we have

With the abbreviation.

This theorem, the divergence is interpreted as a so-called source density of the vector field. Due to the indices or the operator, the dimension of the respective integration manifold is additionally emphasized.

In explicit use of multiple integrals ( waiving the indexing ) for:

Ie: the integral of the divergence of the entire volume is equal to the integral of the flow from the surface.

Second, the Stokes' theorem, which is a statement of differential geometry and can write directly to multiple integrals in the special case of three-dimensional space.

Is a two-dimensional submanifold of the three-dimensional Euclidean space, it shall

Wherein the rotation of the vector field respectively.

This set the rotation of a vector field is interpreted as a so-called vortex density of the vector field; here is the three-component vector and the edge of a closed curve in.

Integration of vector-valued functions

The integration of functions that are not real - or complex-valued, but assume values ​​in a more general vector space, is also possible in many different ways.

The most direct generalization of the Lebesgue integral of Banach space - valued functions is the Bochner integral ( after Salomon Bochner ). Many results of the one-dimensional theory transferred thereby literally on Banach spaces.

The definition of the Riemann integral to transfer to vector-valued functions using Riemann sums, is not difficult. A key difference here, however, that no longer have any Riemann - integrable function is Bochner - integrable.

A common generalization of Bochner and Riemann integral, which resolves this deficiency, the McShane integral, which is most easily defined using generalized Riemann sums.

Also the Birkhoff integral is a common generalization of the Bochner and the Riemann integral. In contrast to the McShane integral, however, the definition of the Birkhoff integral requires no topological structure in the domain of the functions. However, if the conditions for McShane integration met, each Birkhoff integrable function is McShane integrable.

Moreover, even the Pettis integral worth mentioning as the next generalization step. It uses a functional analytic definition, in which the integration is attributed to the one-dimensional case: Let this be a measure space. A function is called while Pettis - integrable if for every continuous functional, the function is Lebesgue integrable and for any measurable quantity a vector exists, so

Applies. The vector is then appropriately designated.

For functions that take values ​​in a separable Banach space V, the Pettis integral coincides with the McShane and the Bochner integral. The most important special case of all these definitions is the case of functions in which can easily be integrated in all these definitions, component by component.

Generalizations

Measure theory

Hair cal measure

The Haar measure by Alfréd hair, represents a generalization of the Lebesgue measure on locally compact topological groups and thus induces an integral as a generalization of the Lebesgue integral.

Integration on manifolds

See: differential form

Finally, integration can also be used to measure surfaces of given bodies. This results in the field of differential geometry.