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Riemann sum

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

Contents

Definition

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})

Related Terms

The Upper sum

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})

The Lower sum

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})

See Also


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