Difference between revisions of "Riemann sum"

Revision as of 11:06, 7 May 2008

A Riemann sum is a finite approximation to the Riemann Integral.

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $\mathcal{\dot{P}}=\{([x_{i-1},x_i],t_i)\}_{i=1}^n$ be a tagged partition on $[a,b]$

The Riemann sum of $f$ with respect to $\mathcal{\dot{P}}$ on $[a,b]$ is defined as $S(f,\mathcal{\dot{P}})=\sum_{i=1}^n f(t_i)(x_i-x_{i-1})$

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n$ be a partition on $[a,b]$

Let $M_i=\sup \{f(x):x\in [x_{i-1},x_i]\}\forall i$

The Upper sum of $f$ with respect to $\mathcal{P}$ on $[a,b]$ is defined as $U(f,\mathcal{P})=\sum_{i=1}^n M_i (x_i-x_{i-1})$

Let $f:[a,b]\rightarrow\mathbb{R}$

Let $\mathcal{P}=\{[x_{i-1},x_i]\}_{i=1}^n$ be a partition on $[a,b]$

Let $m_i=\inf \{f(x):x\in [x_{i-1},x_i]\}\forall i$

The Lower sum of $f$ with respect to $\mathcal{P}$ on $[a,b]$ is defined as $L(f,\mathcal{P})=\sum_{i=1}^n m_i (x_i-x_{i-1})$

@@ Line 1: / Line 1: @@
-A '''Reimann sum''' is a finite approximation to the [[Integral|Reimann Integral]]
+A '''Riemann sum''' is a finite approximation to the [[Integral|Riemann Integral]].
 ==Definition==
@@ Line 6: / Line 6: @@
 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 '''Reimann 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>
+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 Tems==
+==Related Terms==
 ===The Upper sum===
 Let <math>f:[a,b]\rightarrow\mathbb{R}</math>
@@ Line 29: / Line 29: @@
 ==See Also==
 *[[Integral]]
-*[[Partition of an interval|Partition]]
+*[[Partition of an interval]]
 {{stub}}