Source / pinned edition
The Fundamental Theorem of Calculus
Source of package openstax.calculus_integrals · OpenStax Calculus, ch. 5 Integration, module m53632
- work
- urn:openstax:clir:calculus-integrals#M53632
- number
- OpenStax Calculus, ch. 5 Integration, module m53632
- kind
- urn:openstax:clir:calculus-integrals#treatise
m53632/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
- edition
- urn:openstax:clir:calculus-integrals#M53632_MAIN
- language
- en
- officiality
- official
- materialisation
- PINNED_OFFICIAL_BYTES
- life cycle
- adopted: 2026-07-15
Publications
| URI | Type | documentHash | Retrieved |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53632/index.cnxml | application/xml; charset=utf-8 | sha256:730f40d14f0ae8e84dca2c53116a1b044f6521b54e4a4b21f22a500825e33c7a | 2026-09-12T00:00:00Z |
Fragments — 2
module/m53632#fs-id1170571704350 · section
en · official · sha256:02316dab595d2b5496db2054d7a754cd69790a3c8c851ee746f86612fd6a2f82
<note id="fs-id1170571704350" class="theorem">
<title>Fundamental Theorem of Calculus, Part 1</title>
<para id="fs-id1170571679188">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 over 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> and 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 defined by</para>
<equation id="fs-id1170572552076"><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:mo>=</m:mo><m:mstyle displaystyle="true"><m:mrow><m:msubsup><m:mo>∫</m:mo><m:mi>a</m:mi><m:mi>x</m:mi></m:msubsup><m:mrow><m:mi>f</m:mi><m:mrow><m:mo>(</m:mo><m:mi>t</m:mi><m:mo>)</m:mo></m:mrow><m:mi>d</m:mi><m:mi>t</m:mi></m:mrow></m:mrow></m:mstyle><m:mo>,</m:mo></m:mrow></m:math></equation>
<para id="fs-id1170572307182">then <m:math><m:mrow><m:msup><m:mi>F</m:mi><m:mo>′</m:mo></m:msup><m:mrow><m:mtext>(</m:mtext><m:mi>x</m:mi><m:mo>)</m:mo></m:mrow><m:mo>=</m:mo><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> over <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/m53632#fs-id1170571660076 · section
en · official · sha256:6f4fd5054cfa771b335b1a3908a0ba557bd830e426b001fe342d0e589406ae8e
<note id="fs-id1170571660076" class="theorem">
<title>The Fundamental Theorem of Calculus, Part 2</title>
<para id="fs-id1170572448137">If <emphasis effect="italics">f</emphasis> is continuous over the 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:mrow></m:math> and <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 any antiderivative of <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:mo>,</m:mo></m:mrow></m:math> then</para>
<equation id="fs-id1170572230004"><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:mi>F</m:mi><m:mrow><m:mo>(</m:mo><m:mi>b</m:mi><m:mo>)</m:mo></m:mrow><m:mo>−</m:mo><m:mi>F</m:mi><m:mrow><m:mo>(</m:mo><m:mi>a</m:mi><m:mo>)</m:mo></m:mrow><m:mo>.</m:mo></m:mrow></m:math></equation>
</note>The hash beside the text is the sha256 of its exact bytes. It is the check that the text shown matches the one pinned in the package: a divergence fails the oracle’s static analysis.