Source / pinned edition
Integration Formulas and the Net Change Theorem
Source of package openstax.calculus_integrals · OpenStax Calculus, ch. 5 Integration, module m53633
- work
- urn:openstax:clir:calculus-integrals#M53633
- number
- OpenStax Calculus, ch. 5 Integration, module m53633
- kind
- urn:openstax:clir:calculus-integrals#treatise
m53633/index.cnxml at commit 8dbc2ce (main, 2026-07-15)
- edition
- urn:openstax:clir:calculus-integrals#M53633_MAIN
- language
- en
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- official
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- adopted: 2026-07-15
Publications
| URI | Type | documentHash | Retrieved |
|---|---|---|---|
| https://raw.githubusercontent.com/openstax/osbooks-calculus-bundle/8dbc2ce19e804924b2517b89ac72ee45be949d15/modules/m53633/index.cnxml | application/xml; charset=utf-8 | sha256:a94ea4a8f62d8613744d11e9506517dcaf265286f13c5581479a31d5440e6722 | 2026-09-12T00:00:00Z |
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module/m53633#fs-id1170572295126 · section · referenced by: 2
en · official · sha256:b8310d6765be27d37f5121ef8aff7347756fe94bbd3356dc4851bb957ff5eb70
<note id="fs-id1170572295126" class="theorem">
<title>Net Change Theorem</title>
<para id="fs-id1170572480924">The new value of a changing quantity equals the initial value plus the integral of the rate of change:</para>
<equation id="fs-id1170572449540"><m:math><m:mrow><m:mtable><m:mtr/><m:mtr/><m:mtr><m:mtd columnalign="left"><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: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:mo>′</m:mo><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:mtd></m:mtr><m:mtr><m:mtd columnalign="center"><m:mtext>or</m:mtext></m:mtd></m:mtr><m:mtr><m:mtd columnalign="left"><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:mo>′</m:mo><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: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:mrow></m:mrow></m:mstyle><m:mo>.</m:mo></m:mtd></m:mtr></m:mtable></m:mrow></m:math></equation>
</note>Referencing norms: NetChangeJustifiesDefiniteIntegral, definite_integral_justified
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