Difference between revisions of "Riemann sum"
(New page: A '''Reimann sum''' is a finite approximation to the Reimann Integral ==Definition== Let <math>f:[a,b]\rightarro\mathbb{R}</math> Let <math>\mathcal{\dot{P}}=\{([x_{i-1},x_i...) |
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| − | A ''' | + | A '''Riemann sum''' is a finite approximation to the [[Integral|Riemann Integral]]. |
==Definition== | ==Definition== | ||
| − | Let <math>f:[a,b]\ | + | Let <math>f:[a,b]\rightarrow\mathbb{R}</math> |
Let <math>\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n</math> be a [[Partition of an interval|tagged partition]] on <math>[a,b]</math> | Let <math>\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n</math> be a [[Partition of an interval|tagged partition]] on <math>[a,b]</math> | ||
| − | The ''' | + | The '''Riemann sum''' of <math>f</math> with respect to <math>\mathcal{\dot{P}}</math> on <math>[a,b]</math> is defined as <math>S(f,\mathcal{\dot{P}})=\sum_{i=1}^n f(t_i)(x_i-x_{i-1})</math> |
| − | ==Related | + | ==Related Terms== |
===The Upper sum=== | ===The Upper sum=== | ||
| − | Let <math>f:[a,b]\ | + | Let <math>f:[a,b]\rightarrow\mathbb{R}</math> |
Let <math>\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n</math> be a [[Partition of an interval|partition]] on <math>[a,b]</math> | Let <math>\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n</math> be a [[Partition of an interval|partition]] on <math>[a,b]</math> | ||
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Let <math>M_i=\sup \{f(x):x\in [x_{i-1},x_i]\}\forall i</math> | Let <math>M_i=\sup \{f(x):x\in [x_{i-1},x_i]\}\forall i</math> | ||
| − | The '''Upper sum''' of <math>f</math> with respect to <math>\mathcal{P}</math> on <math>[a,b]</math> is defined as <math>U(f,\mathcal{P})= | + | The '''Upper sum''' of <math>f</math> with respect to <math>\mathcal{P}</math> on <math>[a,b]</math> is defined as <math>U(f,\mathcal{P})=\sum_{i=1}^n M_i (x_i-x_{i-1})</math> |
===The Lower sum=== | ===The Lower sum=== | ||
| − | Let <math>f:[a,b]\ | + | Let <math>f:[a,b]\rightarrow\mathbb{R}</math> |
Let <math>\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n</math> be a [[Partition of an interval|partition]] on <math>[a,b]</math> | Let <math>\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n</math> be a [[Partition of an interval|partition]] on <math>[a,b]</math> | ||
| Line 25: | Line 25: | ||
Let <math>m_i=\inf \{f(x):x\in [x_{i-1},x_i]\}\forall i</math> | Let <math>m_i=\inf \{f(x):x\in [x_{i-1},x_i]\}\forall i</math> | ||
| − | The '''Lower sum''' of <math>f</math> with respect to <math>\mathcal{P}</math> on <math>[a,b]</math> is defined as <math>L(f,\mathcal{P})= | + | The '''Lower sum''' of <math>f</math> with respect to <math>\mathcal{P}</math> on <math>[a,b]</math> is defined as <math>L(f,\mathcal{P})=\sum_{i=1}^n m_i (x_i-x_{i-1})</math> |
==See Also== | ==See Also== | ||
*[[Integral]] | *[[Integral]] | ||
| − | *[[Partition of an interval | + | *[[Partition of an interval]] |
{{stub}} | {{stub}} | ||
Latest revision as of 11:06, 7 May 2008
A Riemann sum is a finite approximation to the Riemann Integral.
Definition
Let ![$f:[a,b]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/6/7/c/67c320754a36aaec178e61f04bfcbac41c0bf675.png)
Let ![$\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n$](http://latex.artofproblemsolving.com/1/e/4/1e4a621081b8c5c6b18dfd7163da5adfe7dcb1a2.png)
be a tagged partition on ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
The Riemann sum of 
with respect to 
on is defined as 
Related Terms
The Upper sum
Let
Let ![$\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n$](http://latex.artofproblemsolving.com/8/b/9/8b9f0a714dd3dcc08115d8b1358997a7a8528354.png)
be a partition on
Let ![$M_i=\sup \{f(x):x\in [x_{i-1},x_i]\}\forall i$](http://latex.artofproblemsolving.com/2/a/b/2ab5a8f2754e53b041dd11ce7d0804f6f2c82909.png)
The Upper sum of with respect to 
on is defined as 
The Lower sum
Let
Let be a partition on
Let ![$m_i=\inf \{f(x):x\in [x_{i-1},x_i]\}\forall i$](http://latex.artofproblemsolving.com/a/5/b/a5b6c98e5d3ba94be39aa33425a15155ee1c3b83.png)
The Lower sum of with respect to on is defined as 
See Also
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