Difference between revisions of "The Additivity Theorem"
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| − | '''The Additivity Theorem for Riemann Integrable Functions''': Let <span class="math-inline"><math> f </math></span> be a real-valued function on the interval <span class="math-inline"><math> [a,b] </math></span>, and let <span class="math-inline"><math> c\in (a,b) </math></span>. Then, <span class="math-inline"><math> f </math></span> is Riemann integrable on <span class="math-inline"><math> [a,b] </math></span> if and only if it is also Riemann integrable on <span class="math-inline"><math> [a,c] </math></span> and <span class="math-inline"><math> [c,b] </math></span>. In this case, | + | '''The Additivity Theorem for Riemann Integrable Functions''': Let <span class="math-inline"><math> f </math></span> be a real-valued function on the interval <span class="math-inline"><math> [a,b] </math></span>, and let <span class="math-inline"><math> c\in (a,b) </math></span>. Then, <span class="math-inline"><math> f </math></span> is Riemann integrable on <span class="math-inline"><math> [a,b] </math></span> if and only if it is also Riemann integrable on <span class="math-inline"><math> [a,c] </math></span> and <span class="math-inline"><math> [c,b] </math></span>. In this case, |
| − | <math> | + | <math> \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f </math> |
</blockquote> | </blockquote> | ||
Revision as of 15:46, 9 November 2021
The Additivity Theorem for Riemann Integrable Functions: Let be a real-valued function on the interval , and let . Then, is Riemann integrable on if and only if it is also Riemann integrable on and . In this case,
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle [a,c]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70b45aa26fb924cdd37786af1adcf5d58cfe15af)
![{\displaystyle [c,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07f3f774df12e18b9f2bb94333ff320bbd2c308d)
