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{{terjemah|en|Integral}}
{{proses|BukanTeamBiasa}}
[[Fail:Integral example.svg|thumb|Kamiran tentu suatu fungsi boleh diwakilkan sebagai luas bertanda kawasan yang dibatasi oleh grafnya]]
{{two other uses|the concept of integrals in [[calculus]]|the set of numbers|integer}}
[[File:Integral example.svg|thumb|A definite integral of a function can be represented as the signed area of the region bounded by its graph.]]
{{Calculus|cTopic=Integration}}
'''IntegrationKamiran''' isialah ansatu importantkonsep conceptpenting indalam [[mathematicsmatematik]] whichyang, togetherbersama withdengan [[derivative|differentiationpembezaan]], forms one ofmembentuk theantara mainoperasi operationsutama indalam [[calculuskalkulus]]. Given aDiberi [[functionfungsi (mathematicsmatematik)|functionfungsi]] ''ƒ'' of asatu [[Realpemboleh number|realubah]] [[variablenombor (mathematics)nyata|variablenyata]] ''x'' and andan [[intervalsela (mathematicsmatematik)|intervalsela]] <nowiki>[</nowiki>''a'',&nbsp;''b''<nowiki>]</nowiki> of the [[realgaris linenyata]], the '''definitekamiran integraltentu'''
 
: <math>\int_a^b f(x)\,dx \, ,</math>
 
isditakrifkan definedsecara informallytidak toformal be the net signedsebagai [[area (geometry)keluasan|arealuas]] ofbertanda thebersih regionkawasan in thedi satah-''xy''-plane boundedyang bydibatasi thedengan [[Graphgraf of a functionfungsi|graphgraf]] of ''ƒ'', the paksi-''x''-axis, and thedan verticalgaris linesmenegak ''x''&nbsp;= ''a'' anddan ''x''&nbsp;=&nbsp;''b''.
 
Istilah ''kamiran'' juga boleh merujuk kepada tanggapan [[antiterbitan]], fungsi ''F'' yang [[pembezaan|terbitan]]nya ialah fungsi diberi ''ƒ''. Dalam kes ini ia dipanggil ''kamiran tak tentu'', manakala kamiran yang dibincangkan dalam rencana ini dipanggil ''kamiran tentu''. Sesetengah penulis mengekalkan perbezaan antara antiterbitan dan kamiran tak tentu.
The term ''integral'' may also refer to the notion of [[antiderivative]], a function ''F'' whose [[derivative]] is the given function ''ƒ''. In this case it is called an ''indefinite integral'', while the integrals discussed in this article are termed ''definite integrals''. Some authors maintain a distinction between antiderivatives and indefinite 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 [[differential calculus|differentiation]]: if ''ƒ'' is a continuous real-valued function defined on a [[closed interval]] [''a'',&nbsp;''b''], then, once an antiderivative ''F'' of ''ƒ'' is known, the definite integral of ''ƒ'' over that interval is given by
Baris 94 ⟶ 93:
 
=== Riemann integral ===
{{mainutama|Riemann integral}}
[[Image:Integral Riemann sum.png|thumb|right|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 sum]]s of functions with respect to ''tagged partitions'' of an interval. Let [''a'',''b''] be a [[Interval (mathematics)|closed interval]] of the real line; then a ''tagged partition'' of [''a'',''b''] is a finite sequence
Baris 109 ⟶ 108:
 
=== Lebesgue integral ===
{{mainutama|Lebesgue integration}}
 
The Riemann integral is not defined for a wide range of functions and situations of importance in applications (and of interest in theory). For example, the Riemann integral can easily integrate density to find the mass of a steel beam, but cannot accommodate a steel ball resting on it. This motivates other definitions, under which a broader assortment of functions is integrable {{Harv|Rudin|1987}}. The Lebesgue integral, in particular, achieves great flexibility by directing attention to the weights in the weighted sum.
Baris 246 ⟶ 245:
Instead of viewing the above as conventions, one can also adopt the point of view that integration is performed on [[Orientability|''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 (see below for integration of differential forms):
: <math>\int_M \omega = - \int_{M'} \omega \,.</math>
<!--
 
==Fundamental theorem of calculus==
{{mainutama|Fundamental theorem of calculus}}
 
The ''fundamental theorem of calculus'' is the statement that [[derivative|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.
Baris 264 ⟶ 263:
:is an anti-derivative of ''f'' on [''a'', ''b'']. Moreover,
::<math>\int_a^b f(t) \, dt = F(b) - F(a).</math>
 
== Extensions ==
=== Improper integrals ===
{{main|Improper integral}}
[[Image:Improper integral.svg|right|thumb|The [[improper integral]]<br /><math>\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} = \pi</math><br /> has unbounded intervals for both domain and range.]]
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 (mathematics)|limit]] of a [[sequence]] of proper [[Riemann integral]]s 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.
:<math>\int_{a}^{\infty} f(x)dx = \lim_{b \to \infty} \int_{a}^{b} f(x)dx</math>
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.
:<math>\int_{a}^{b} f(x)dx = \lim_{\epsilon \to 0} \int_{a+\epsilon}^{b} f(x)dx</math>
 
That is, the improper integral is the [[limit (mathematics)|limit]] of proper integrals as one endpoint of the interval of integration approaches either a specified [[real number]], or ∞, or &minus;∞. In more complicated cases, limits are required at both endpoints, or at interior points.
 
Consider, for example, the function <math>\tfrac{1}{(x+1)\sqrt{x}}</math> 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 <math>\tfrac{\pi}{6}</math>. To integrate from 1 to ∞, a Riemann sum is not possible. However, any finite upper bound, say ''t'' (with ''t''&nbsp;&gt;&nbsp;1), gives a well-defined result, <math>\tfrac{\pi}{2} - 2\arctan \tfrac{1}{\sqrt{t}}</math>. This has a finite limit as ''t'' goes to infinity, namely <math>\tfrac{\pi}{2}</math>. Similarly, the integral from <sup>1</sup>⁄<sub>3</sub> to 1 allows a Riemann sum as well, coincidentally again producing <math>\tfrac{\pi}{6}</math>. Replacing <sup>1</sup>⁄<sub>3</sub> by an arbitrary positive value ''s'' (with ''s''&nbsp;&lt;&nbsp;1) is equally safe, giving <math>-\tfrac{\pi}{2} + 2\arctan\tfrac{1}{\sqrt{s}}</math>. This, too, has a finite limit as ''s'' goes to zero, namely <math>\tfrac{\pi}{2}</math>. Combining the limits of the two fragments, the result of this improper integral is
:<math>\begin{align}
\int_{0}^{\infty} \frac{dx}{(x+1)\sqrt{x}} &{} = \lim_{s \to 0} \int_{s}^{1} \frac{dx}{(x+1)\sqrt{x}}
+ \lim_{t \to \infty} \int_{1}^{t} \frac{dx}{(x+1)\sqrt{x}} \\
&{} = \lim_{s \to 0} \left( - \frac{\pi}{2} + 2 \arctan\frac{1}{\sqrt{s}} \right)
+ \lim_{t \to \infty} \left( \frac{\pi}{2} - 2 \arctan\frac{1}{\sqrt{t}} \right) \\
&{} = \frac{\pi}{2} + \frac{\pi}{2} \\
&{} = \pi .
\end{align}</math>
This process is not guaranteed success; a limit may fail to exist, or may be unbounded. For example, over the bounded interval 0 to 1 the integral of <math>\tfrac{1}{x}</math> does not converge; and over the unbounded interval 1 to ∞ the integral of <math>\tfrac{1}{\sqrt{x}}</math> does not converge.
 
<!-- [[Image:Improper integral unbounded internally.svg|right|thumb|The [[improper integral]]<br /><math>\int_{-1}^{1} \frac{dx}{\sqrt[3]{x^2}} = 6</math><br /> is unbounded internally, but both left and right limits exist.]] -->
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
:<math>\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}</math>
But the similar integral
:<math> \int_{-1}^{1} \frac{dx}{x} \,\!</math>
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 ===
{{main article|Multiple integral}}
[[Image:Volume under surface.png|right|thumb|Double integral as volume under a surface.]]
Integrals can be taken over regions other than intervals. In general, an integral over a [[Set (mathematics)|set]] ''E'' of a function ''f'' is written:
 
:<math>\int_E f(x) \, dx.</math>
 
Here ''x'' need not be a real number, but can be another suitable quantity, for instance, a [[Vector (geometric)|vector]] in '''R'''<sup>3</sup>. [[Fubini's theorem]] shows that such integrals can be rewritten as an ''[[Multiple integral|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 (mathematics)|domain]]. (The same volume can be obtained via the ''triple integral'' &mdash; the integral of a function in three variables &mdash; 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 [[Fourth dimension|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 &times; 6 &times; 5 may be obtained in two ways:
* By the double integral
:: <math>\iint_D 5 \ dx\, dy</math>
: 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 2 ≤ ''x'' ≤ 7, 4 ≤ ''y'' ≤ 9, our above double integral now reads
 
::<math>\int_4^9 \int_2^7 \ 5 \ dx\, dy</math>
 
: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 <math>F(b) - F(a)</math> method or otherwise, the result is again integrated with respect to the other variable. The result will equate to the volume under the surface.
 
* By the triple integral
::<math>\iiint_\mathrm{cuboid} 1 \, dx\, dy\, dz</math>
:of the constant function 1 calculated on the cuboid itself.
 
=== Line integrals ===
{{main|Line integral}}
[[Image:Line-Integral.gif|right|thumb|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 field]]s.
 
A ''line integral'' (sometimes called a ''path integral'') is an integral where the [[function (mathematics)|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 [[Inner product space|scalar product]] of the vector field with a [[Differential (infinitesimal)|differential]] vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on [[interval (mathematics)|interval]]s. Many simple formulas in physics have natural continuous analogs in terms of line integrals; for example, the fact that [[Mechanical work|work]] is equal to [[force]], ''F'', multiplied by displacement, ''s'', may be expressed (in terms of vector quantities) as:
:<math>W=\vec F\cdot\vec s.</math>
For an object moving along a path in a [[vector field]] <math>\vec F</math> 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 <math>\vec s</math> to <math>\vec s + d\vec s</math>. This gives the line integral
:<math>W=\int_C \vec F\cdot d\vec s.</math>
 
=== Surface integrals ===
{{main|Surface integral}}
[[Image:Surface integral illustration.png|right|thumb|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 [[Multiple integral|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:
:<math>\int_S {\mathbf v}\cdot \,d{\mathbf {S}}.</math>
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 ===
{{main|differential form}}
 
A [[differential form]] is a mathematical concept in the fields of [[multivariable calculus]], [[differential topology]] and [[tensor]]s. The modern notation for the differential form, as well as the idea of the differential forms as being the [[Exterior algebra|wedge products]] of [[exterior derivative]]s forming an [[exterior algebra]], was introduced by [[Élie Cartan]].
 
We initially work in an [[open set]] in '''R'''<sup>''n''</sup>.
A 0-form is defined to be a [[smooth function]] ''f''.
When we integrate a [[function (mathematics)|function]] ''f'' over an ''m''-[[dimension]]al subspace ''S'' of '''R'''<sup>''n''</sup>, we write it as
:<math>\int_S f\,dx^1 \cdots dx^m.</math>
 
(The superscripts are indices, not exponents.) We can consider ''dx''<sup>1</sup> through ''dx''<sup>''n''</sup> to be formal objects themselves, rather than tags appended to make integrals look like [[Riemann sum]]s. Alternatively, we can view them as [[One-form|covectors]], and thus a [[measure (mathematics)|measure]] of "density" (hence integrable in a general sense). We call the ''dx''<sup>1</sup>, …,''dx<sup>n</sup>'' ''basic'' [[one-form|1-''forms'']].
 
We define the [[Exterior algebra|wedge product]], "∧", a bilinear "multiplication" operator on these elements, with the ''alternating'' property that
 
:<math> dx^a \wedge dx^a = 0 \,\!</math>
 
for all indices ''a''. Note that alternation along with linearity implies ''dx''<sup>''b''</sup>∧''dx''<sup>''a''</sup>&nbsp;= −''dx''<sup>''a''</sup>∧''dx''<sup>''b''</sup>. This also ensures that the result of the wedge product has an [[Orientation (mathematics)|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 ''dx''<sup>''a''</sup>∧''dx''<sup>''b''</sup>∧''dx''<sup>''c''</sup> 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 '''R'''<sup>''n''</sup> at most ''n'' covectors can be linearly independent, thus a ''k-''form with ''k''&nbsp;&gt;&nbsp;''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'' ''dx<sup>a</sup>'' over '''R'''<sup>''n''</sup>, we define the action of '''d''' by:
 
:<math>{\bold d}{\omega} = \sum_{i=1}^n \frac{\partial f}{\partial x_i} dx^i \wedge dx^a.</math>
 
with extension to general ''k''-forms occurring linearly.
 
This more general approach allows for a more natural coordinate-free approach to integration on [[manifold]]s. It also allows for a natural generalisation of the [[fundamental theorem of calculus]], called [[Stokes' theorem]], which we may state as
 
:<math>\int_{\Omega} {\bold d}\omega = \int_{\partial\Omega} \omega \,\!</math>
 
where ω is a general ''k''-form, and ∂Ω denotes the [[boundary (topology)|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 dual]]ity, 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.
 
== Methods ==
Baris 408 ⟶ 293:
 
=== Symbolic algorithms ===
{{mainutama|Symbolic integration}}
 
Many problems in mathematics, physics, and engineering involve integration where an explicit formula for the integral is desired. Extensive [[Lists of integrals|tables of integrals]] have been compiled and published over the years for this purpose. With the spread of [[computer]]s, many professionals, educators, and students have turned to [[computer algebra system]]s that are specifically designed to perform difficult or tedious tasks, including integration. Symbolic integration presents a special challenge in the development of such systems.
Baris 419 ⟶ 304:
 
=== Numerical quadrature ===
{{mainutama|numerical integration}}
 
The integrals encountered in a basic calculus course are deliberately chosen for simplicity; those found in real applications are not always so accommodating. Some integrals cannot be found exactly, some require special functions which themselves are a challenge to compute, and others are so complex that finding the exact answer is too slow. This motivates the study and application of numerical methods for approximating integrals, which today use [[Floating point|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.
Baris 443 ⟶ 328:
| colspan="2" | &nbsp;2.33041 || colspan="2" | &nbsp;2.58562 || colspan="2" | &nbsp;2.62934 || colspan="2" | &nbsp;1.64019 || colspan="2" | −0.32444 || colspan="2" | −1.09159 || colspan="2" | −0.60387 || colspan="2" | &nbsp;0.31734 ||
|- style="background-color:#aaa"
| || || || || || || || || || || || || || || || || || || <!-- extra row improves column spacing -->
|}
[[Image:Numerical quadrature 4up.png|thumb|right|Numerical quadrature methods: <span style="color:#bc1e47">■</span>&nbsp;Rectangle, <span style="color:#fec200">■</span>&nbsp;Trapezoid, <span style="color:#0081cd">■</span>&nbsp;Romberg, <span style="color:#009246">■</span>&nbsp;Gauss]]
Baris 477 ⟶ 362:
 
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.
-->
 
== SeeLihat alsojuga ==
{{portalpar|MathematicsMatematik}}
<div style="-moz-column-count:2; column-count:2;">
* [[Lists of integrals]] – integrals of the most common functions
Baris 495 ⟶ 380:
</div>
 
==Notes Nota ==
{{reflist}}
 
==References Rujukan ==
* {{citation | last=Apostol | first=Tom M. | author-link=Tom M. Apostol | title=Calculus, Vol.&nbsp;1: One-Variable Calculus with an Introduction to Linear Algebra | year=1967 | edition=2nd | publisher=[[John Wiley & Sons|Wiley]] | isbn=978-0-471-00005-1}}
* {{citation | last=Bourbaki| first=Nicolas | author-link=Nicolas Bourbaki | title=Integration I | year=2004 | publisher=[[Springer Science+Business Media|Springer Verlag]] | isbn=3-540-41129-1}}. In particular chapters III and IV.
Baris 522 ⟶ 407:
* {{citation | author=W3C | year=2006<!--January--> | title=Arabic mathematical notation<!--W3C Interest Group Note 31--> | url=http://www.w3.org/TR/arabic-math/}}
 
== Pautan luar ==
==External links==
{{wikibooks|CalculusKalkulus}}
* [http://integrals.wolfram.com/ The Integrator] by [[Wolfram Research]]
* [http://mathworld.wolfram.com/RiemannSum.html Riemann Sum] by [[Wolfram Research]]
Baris 529 ⟶ 414:
* [http://user.mendelu.cz/marik/maw/index.php?lang=en&form=integral Mathematical Assistant on Web] online calculation of integrals, allows to integrate in small steps (includes also hints for next step which cover techniques like by parts, substitution, partial fractions, application of formulas and others, powered by [[Maxima (software)]])
 
===OnlineBuku booksdalam talian===
* Keisler, H. Jerome, [http://www.math.wisc.edu/~keisler/calc.html Elementary Calculus: An Approach Using Infinitesimals], University of Wisconsin
* Stroyan, K.D., [http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/InfsmlCalc.htm A Brief Introduction to Infinitesimal Calculus], University of Iowa
Baris 543 ⟶ 428:
{{integral}}
 
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