Riemann sum

A Reimann sum is a finite approximation to the Reimann Integral

Definition

Let $f:[a,b]\rightarro\mathbb{R}$ (Error compiling LaTeX. Unknown error_msg)

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

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

Related Tems

The Upper sum

Let $f:[a,b]\rightarro\mathbb{R}$ (Error compiling LaTeX. Unknown error_msg)

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 $U(f,\mathcal{P})=\displaystyle\sum_{i=1}^n M_i (x_i-x_{i-1})$

The Lower sum

Let $f:[a,b]\rightarro\mathbb{R}$ (Error compiling LaTeX. Unknown error_msg)

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 $L(f,\mathcal{P})=\displaystyle\sum_{i=1}^n m_i (x_i-x_{i-1})$

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Riemann sum

Contents

Definition

Related Tems

The Upper sum

The Lower sum

See Also