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> \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f </math> | + | <div style="text-align: center;"> <math> \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f </math> </div> |
</blockquote> | </blockquote> | ||
Revision as of 15:48, 9 November 2021
The Additivity Theorem for Riemann Integrable Functions: Let be a real-valued function on the interval , and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\in (a,b) } . Then, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } is Riemann integrable on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b] } if and only if it is also Riemann integrable on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,c] } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [c,b] } . In this case,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int_{a}^{b} f = \int_{a}^{c} f + \int_{c}^{b} f }
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
