Integral: Perbedaan antara revisi
Konten dihapus Konten ditambahkan
Tag: Suntingan perangkat seluler Suntingan peramban seluler Suntingan seluler lanjutan |
Tag: Suntingan perangkat seluler Suntingan peramban seluler Suntingan seluler lanjutan |
||
Baris 255:
* Integral [[jalur kasar]], yang ditentukan untuk fungsi yang dilengkapi dengan beberapa "jalur kasar" tambahan menyusun dan menggeneralisasi integrasi stokastik terhadap [[semi pesergi panjang]] dan proses seperti [[gerakan pecahan Brownian]].
* [[Choquet integral]], integral subaditif atau superaditif yang dibuat oleh ahli matematika Prancis Gustave Choquet pada tahun 1953.
== Properti ==
=== Linearitas ===
Kumpulan fungsi yang dapat diintegrasikan Riemann pada interval tertutup {{math|[''a'', ''b'']}} membentuk [[ruang vektor]] di bawah operasi [[penambahan pointwise]] dan perkalian dengan skalar, dan operasi integral
:<math> f \mapsto \int_a^b f(x) \; dx</math>
<!--- mubazir
untuk integral [[fungsi (matematika)|fungsi]] {{mvar|f}} pada {{math|[''a'', ''b'']}}
--->
adalah [[fungsional linear]] pada ruang vektor ini. Jadi, pertama, kumpulan dari fungsi terintegral ditutup pada pengambilan [[kombinasi linier]]; dan kedua, integral dari kombinasi linier adalah kombinasi linier dari integral,<ref name=":2" />
<!--- leftover from the past text; redundant
For example, in Riemann integration, if {{mvar|f}} and {{mvar|g}} are [[real number|real-valued]] integrable functions on a [[closed set|closed]] and [[bounded set|bounded]] [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}}, and {{mvar|α}} and {{mvar|β}} are real numbers, then the function {{math|''αf'' + ''βg''}} defined by {{math|(''αf'' + ''βg'')(''x'') {{=}} ''αf''(''x'') + ''βg''(''x'')}} for all {{mvar|x}} in {{math|[''a'', ''b'']}} is integrable, with
--->
:<math> \int_a^b (\alpha f + \beta g)(x) \, dx = \alpha \int_a^b f(x) \,dx + \beta \int_a^b g(x) \, dx. \,</math>
Demikian pula, himpunan [[bilangan real | nyata]] - nilai fungsi terintegralkan Lebesgue pada [[Ukuran (matematika)|ruang ukur]] yang diberikan {{mvar|E}} dengan ukuran {{mvar|μ}} ditutup dengan mengambil kombinasi linier, dan karenanya membentuk ruang vektor, dan integral Lebesgue
: <math> f\mapsto \int_E f \, d\mu </math>
adalah fungsi linear pada ruang vektor ini, sehingga
:<math> \int_E (\alpha f + \beta g) \, d\mu = \alpha \int_E f \, d\mu + \beta \int_E g \, d\mu. </math>
Secara lebih umum, pertimbangkan ruang vektor dari semua [[fungsi terukur]] pada ruang ukur {{math|(''E'',''μ'')}}, mengambil nilai dalam [[ruang kompak lokal|kompak lokal]] [[spasi metrik lengkap|lengkap]] [[spasi vektor topologi]] {{mvar|V}} di atas [[gelanggang topologi|bidang topologi]] {{math|''K'', ''f'' : ''E'' → ''V''}}. Kemudian seseorang dapat mendefinisikan peta integrasi abstrak yang ditugaskan ke setiap fungsi {{mvar|f}} sebuah elemen dari {{mvar|V}} atau simbol {{math|''∞''}},
:<math> f\mapsto\int_E f \,d\mu, \,</math>
kompatibel dengan kombinasi linear. Dalam situasi ini, linieritas berlaku untuk subruang fungsi yang integralnya merupakan elemen dari {{mvar | V}} (yaitu "finite"). Kasus khusus yang paling penting muncul adalah {{mvar|K}} pada {{math|'''R'''}}, {{math|'''C'''}}, atau perluasan lapangan yang terbatas {{math|'''Q'''<sub>''p''</sub>}} dari [[bilangan p-adic]] s, dan {{mvar|V}} adalah ruang vektor berdimensi-hingga di atas {{mvar|K}}, dan jika {{math|''K'' {{=}} '''C'''}} dan {{mvar|V}} adalah kompleks [[ruang Hilbert]].
Linearitas, bersama dengan beberapa sifat kontinuitas alami dan normalisasi untuk kelas fungsi "sederhana" tertentu, dapat digunakan untuk memberikan definisi alternatif dari integral. Ini adalah pendekatan dari [[Integral Daniell|Daniell]] untuk kasus fungsi bernilai riil pada suatu himpunan {{mvar|X}}, digeneralisasikan oleh [[Nicolas Bourbaki]] ke fungsi dengan nilai dalam ruang vektor topologi yang kompak secara lokal. Lihat {{Harv|Hildebrandt|1953}} untuk karakterisasi aksiomatik dari integral.
<!--=== Ketimpangan ===
A number of general inequalities hold for Riemann-integrable [[function (mathematics)|functions]] defined on a [[closed set|closed]] and [[bounded set|bounded]] [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}} and can be generalized to other notions of integral (Lebesgue and Daniell).
* ''Upper and lower bounds.'' An integrable function {{mvar|f}} on {{math|[''a'', ''b'']}}, is necessarily [[bounded function|bounded]] on that interval. Thus there are [[real number]]s {{mvar|m}} and {{mvar|M}} so that {{math|''m'' ≤ ''f'' (''x'') ≤ ''M''}} for all {{mvar|x}} in {{math|[''a'', ''b'']}}. Since the lower and upper sums of {{mvar|f}} over {{math|[''a'', ''b'']}} are therefore bounded by, respectively, {{math|''m''(''b'' − ''a'')}} and {{math|''M''(''b'' − ''a'')}}, it follows that
:: <math> m(b - a) \leq \int_a^b f(x) \, dx \leq M(b - a). </math>
* ''Inequalities between functions.'' If {{math|''f''(''x'') ≤ ''g''(''x'')}} for each {{mvar|x}} in {{math|[''a'', ''b'']}} then each of the upper and lower sums of {{mvar|f}} is bounded above by the upper and lower sums, respectively, of {{mvar|g}}. Thus
:: <math> \int_a^b f(x) \, dx \leq \int_a^b g(x) \, dx. </math>
:This is a generalization of the above inequalities, as {{math|''M''(''b'' − ''a'')}} is the integral of the constant function with value {{mvar|M}} over {{math|[''a'', ''b'']}}.
:In addition, if the inequality between functions is strict, then the inequality between integrals is also strict. That is, if {{math|''f''(''x'') < ''g''(''x'')}} for each {{mvar|x}} in {{math|[''a'', ''b'']}}, then
:: <math> \int_a^b f(x) \, dx < \int_a^b g(x) \, dx. </math>
* ''Subintervals.'' If {{math|[''c'', ''d'']}} is a subinterval of {{math|[''a'', ''b'']}} and {{math|''f''(''x'')}} is non-negative for all {{mvar|x}}, then
:: <math> \int_c^d f(x) \, dx \leq \int_a^b f(x) \, dx. </math>
* ''Products and absolute values of functions.'' If {{mvar|f}} and {{mvar|g}} are two functions, then we may consider their [[pointwise product]]s and powers, and [[absolute value]]s:
:: <math>
(fg)(x)= f(x) g(x), \; f^2 (x) = (f(x))^2, \; |f| (x) = |f(x)|.\,</math>
:If {{mvar|f}} is Riemann-integrable on {{math|[''a'', ''b'']}} then the same is true for {{math|{{abs|''f''}}}}, and<ref name=":2" />
::<math>\left| \int_a^b f(x) \, dx \right| \leq \int_a^b | f(x) | \, dx. </math>
:Moreover, if {{mvar|f}} and {{mvar|g}} are both Riemann-integrable then {{math|''fg''}} is also Riemann-integrable, and
:: <math>\left( \int_a^b (fg)(x) \, dx \right)^2 \leq \left( \int_a^b f(x)^2 \, dx \right) \left( \int_a^b g(x)^2 \, dx \right). </math>
: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 space|inner product]] of two [[Square-integrable function|square-integrable]] functions {{mvar|f}} and {{mvar|g}} on the interval {{math|[''a'', ''b'']}}.
* ''Hölder's inequality''. Suppose that {{mvar|p}} and {{mvar|q}} are two real numbers, {{math|1 ≤ ''p'', ''q'' ≤ ∞}} with {{math|{{sfrac|1|''p''}} + {{sfrac|1|''q''}} {{=}} 1}}, and {{mvar|f}} and {{mvar|g}} are two Riemann-integrable functions. Then the functions {{math|{{abs|''f''}}<sup>''p''</sup>}} and {{math|{{abs|''g''}}<sup>''q''</sup>}} are also integrable and the following [[Hölder's inequality]] holds:
::<math>\left|\int f(x)g(x)\,dx\right| \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} \left(\int\left|g(x)\right|^q\,dx\right)^{1/q}.</math>
:For {{mvar|p}} = {{mvar|q}} = 2, Hölder's inequality becomes the Cauchy–Schwarz inequality.
* ''Minkowski inequality''. Suppose that {{math|''p'' ≥ 1}} is a real number and {{mvar|f}} and {{mvar|g}} are Riemann-integrable functions. Then {{math|{{abs| ''f'' }}<sup>''p''</sup>, {{abs| ''g'' }}<sup>''p''</sup>}} and {{math|{{abs| ''f'' + ''g'' }}<sup>''p''</sup>}} are also Riemann-integrable and the following [[Minkowski inequality]] holds:
::<math>\left(\int \left|f(x)+g(x)\right|^p\,dx \right)^{1/p} \leq
\left(\int \left|f(x)\right|^p\,dx \right)^{1/p} +
\left(\int \left|g(x)\right|^p\,dx \right)^{1/p}.</math>
: An analogue of this inequality for Lebesgue integral is used in construction of [[Lp space|L<sup>p</sup> spaces]].
===Conventions===
In this section, {{mvar|f}} is a [[real number|real-]]valued Riemann-integrable [[function (mathematics)|function]]. The integral
:<math> \int_a^b f(x) \, dx </math>
over an interval {{math|[''a'', ''b'']}} is defined if {{math|''a'' < ''b''}}. This means that the upper and lower sums of the function {{mvar|f}} are evaluated on a partition {{math|''a'' {{=}} ''x''<sub>0</sub> ≤ ''x''<sub>1</sub> ≤ . . . ≤ ''x''<sub>''n''</sub> {{=}} ''b''}} whose values {{math|''x''<sub>''i''</sub>}} are increasing. Geometrically, this signifies that integration takes place "left to right", evaluating {{mvar|f}} within intervals {{math|[''x''<sub> ''i''</sub> , ''x''<sub> ''i'' +1</sub>]}} where an interval with a higher index lies to the right of one with a lower index. The values {{mvar|a}} and {{mvar|b}}, the end-points of the [[interval (mathematics)|interval]], are called the [[limits of integration]] of {{mvar|f}}. Integrals can also be defined if {{math|''a'' > ''b''}}:
* ''Reversing limits of integration.'' If {{math|''a'' > ''b''}} then define
:: <math>\int_a^b f(x) \, dx = - \int_b^a f(x) \, dx. </math>
This, with {{math|''a'' {{=}} ''b''}}, implies:
* ''Integrals over intervals of length zero.'' If {{mvar|a}} is a [[real number]] then
:: <math>\int_a^a f(x) \, dx = 0. </math>
The first convention is necessary in consideration of taking integrals over subintervals of {{math|[''a'', ''b'']}}; the second says that an integral taken over a degenerate interval, or a [[Point (geometry)|point]], should be [[0 (number)|zero]]. One reason for the first convention is that the integrability of {{mvar|f}} on an interval {{math|[''a'', ''b'']}} implies that {{mvar|f}} is integrable on any subinterval {{math|[''c'', ''d'']}}, but in particular integrals have the property that:
* ''Additivity of integration on intervals.'' If {{mvar|c}} is any [[element (mathematics)|element]] of {{math|[''a'', ''b'']}}, then<ref name=":2" />
:: <math> \int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx.</math>
With the first convention, the resulting relation
: <math>\begin{align}
\int_a^c f(x) \, dx &{}= \int_a^b f(x) \, dx - \int_c^b f(x) \, dx \\
&{} = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx
\end{align}</math>
is then well-defined for any cyclic permutation of {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}.
== Mencari nilai integral ==
| |||