Источник / закреплённая редакция
The Definite Integral
Источник пакета openstax.calculus_integrals · OpenStax Calculus, ch. 5 Integration, module m53631
- work
- urn:openstax:clir:calculus-integrals#M53631
- номер
- OpenStax Calculus, ch. 5 Integration, module m53631
- вид
- urn:openstax:clir:calculus-integrals#treatise
m53631/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
- edition
- urn:openstax:clir:calculus-integrals#M53631_MAIN
- язык
- en
- официальность
- official
- материализация
- PINNED_OFFICIAL_BYTES
- жизненный цикл
- adopted: 2026-07-15
Публикации
| URI | Тип | documentHash | Получено |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53631/index.cnxml | application/xml; charset=utf-8 | sha256:12ffeaa33f9f990dedc128214239abb7fbef92d13556990bb1c79a6ec0ed223e | 2026-09-12T00:00:00Z |
Фрагменты — 2
module/m53631#fs-id1170572103713 · section
en · official · sha256:c2e9cc12800df53ca5145c47f0a29c6d6525800298afb1aa2ad525afb41ff2ecEN
<note id="fs-id1170572103713" class="theorem">
<title>Continuous Functions Are Integrable</title>
<para id="fs-id1170572207625">If <m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> is continuous on <m:math><m:mrow><m:mrow><m:mo>[</m:mo><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:mi>b</m:mi></m:mrow><m:mo>]</m:mo></m:mrow><m:mo>,</m:mo></m:mrow></m:math> then <emphasis effect="italics">f</emphasis> is integrable on <m:math><m:mrow><m:mrow><m:mo>[</m:mo><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:mi>b</m:mi></m:mrow><m:mo>]</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math></para>
</note>module/m53631#fs-id1170572548884 · defn
en · official · sha256:548d6f5059966b136bf5a453e75631f493e2e17d4bb0809fa5a2b9d317607d0aEN
<note id="fs-id1170572548884">
<title>Definition</title>
<para id="fs-id1170572449507">If <m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> is a function defined on an interval <m:math><m:mrow><m:mrow><m:mo>[</m:mo><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:mi>b</m:mi></m:mrow><m:mo>]</m:mo></m:mrow><m:mo>,</m:mo></m:mrow></m:math> the <term id="term-00001">definite integral</term> of <emphasis effect="italics">f</emphasis> from <emphasis effect="italics">a</emphasis> to <emphasis effect="italics">b</emphasis> is given by</para>
<equation id="fs-id1170572280427"><m:math><m:mrow><m:mstyle displaystyle="true"><m:mrow><m:msubsup><m:mo>∫</m:mo><m:mi>a</m:mi><m:mi>b</m:mi></m:msubsup><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mi>d</m:mi><m:mi>x</m:mi></m:mrow></m:mrow></m:mstyle><m:mo>=</m:mo><m:munder><m:mstyle mathsize="140%" displaystyle="true"><m:mrow><m:mtext>lim</m:mtext></m:mrow></m:mstyle><m:mrow><m:mi>n</m:mi><m:mo stretchy="false">→</m:mo><m:mi>∞</m:mi></m:mrow></m:munder><m:mstyle displaystyle="true"><m:munderover><m:mo>∑</m:mo><m:mrow><m:mi>i</m:mi><m:mo>=</m:mo><m:mn>1</m:mn></m:mrow><m:mi>n</m:mi></m:munderover><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mrow><m:msubsup><m:mi>x</m:mi><m:mi>i</m:mi><m:mo>*</m:mo></m:msubsup></m:mrow><m:mo>)</m:mo></m:mrow><m:mtext>Δ</m:mtext><m:mi>x</m:mi></m:mrow></m:mstyle><m:mo>,</m:mo></m:mrow></m:math></equation>
<para id="fs-id1170572342258">provided the limit exists. If this limit exists, the function <m:math><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow></m:mrow></m:math> is said to be integrable on <m:math><m:mrow><m:mrow><m:mo>[</m:mo><m:mrow><m:mi>a</m:mi><m:mo>,</m:mo><m:mi>b</m:mi></m:mrow><m:mo>]</m:mo></m:mrow><m:mo>,</m:mo></m:mrow></m:math> or is an <term id="term-00002">integrable function</term>.</para>
</note>Хэш рядом с текстом — sha256 его точных байтов. Он и есть проверка того, что показанный текст совпадает с закреплённым в пакете: расхождение роняет статику оракула.