Sigma notation

\begin{matrix} \underset{\underset{6}{⏟}}{1 + 1 + 1 + 1 + 1 + 1} & = & \underset{\underset{6}{⏟}}{1 + \dots + 1} = \sum_{κ = 1}^{6} 1 \\ 1 + 2 + 3 + 4 + 5 + 6 & = & 1 + 2 + \dots + 6 = \sum_{κ = 1}^{6} κ \\ 3^{1} + 3^{2} + \dots + 3^{1000} & = & \sum_{κ = 1}^{1000} 3^{κ} \end{matrix}

\begin{matrix} 5^{-2} + 5^{-1} + 5^{0} + 5^{1} + 5^{2} + 5^{3} & = & \sum_{κ = -2}^{3} 5^{κ} \\ sin (\frac{3 π}{2}) + sin (\frac{5 π}{2}) + sin (\frac{7 π}{2}) & = & \sum_{κ = 1}^{3} sin (\frac{(2 κ + 1) π}{2}) \\ = & \sum_{κ = 2}^{4} sin (\frac{(2 κ - 1) π}{2}) \end{matrix}

\sum_{κ = 𝓁}^{𝓊} (α 𝒶_{κ} + β 𝒷_{κ}) = α \sum_{κ = 𝓁}^{𝓊} 𝒶_{κ} + β \sum_{κ = 𝓁}^{𝓊} 𝒷_{κ}

\begin{matrix} \sum_{κ = 1}^{n} κ & = & \frac{n (n + 1)}{2} \\ \sum_{κ = 1}^{n} κ^{2} & = & \frac{n (n + 1) (2 n + 1)}{6} \\ \sum_{κ = 1}^{n} κ^{3} & = & {(\frac{n (n + 1)}{2})}^{2} \end{matrix}

\begin{matrix} \sum_{κ = 1}^{200} {(κ - 3)}^{2} \\ \sum_{κ = 1}^{6} (κ^{3} - κ^{2}) \\ \sum_{κ = 10}^{20} κ \end{matrix}

Area under a curve

Find the area under the curve

f (x) = \frac{1}{32} (x - 1) {(x - 7)}^{2} + 3

Find left end $ℒ_{n}$ and right end $ℛ_{n}$ approximation for

f (x) = \frac{1}{32} (x - 1) {(x - 7)}^{2} + 3

using a regular partition with $n = 5$ on the interval $[1, 11]$

Find upper $𝒰_{n}$ and lower $𝒲_{n}$ approximation for

f (x) = x^{2} + x

using a regular partition with $n$ parts on the interval $[0, 1]$

Riemann Sum

Interval partition

A set of points $𝒫 = {χ_{0}, χ_{1}, χ_{2}, \dots, χ_{n - 1}, χ_{n}}$ with

𝒶 = χ_{0} < χ_{1} < χ_{2} < \dots < χ_{n - 1} < χ_{n} = 𝒷

which divides the interval $[𝒶, 𝒷]$ into subintervals of the form

[χ_{0}, χ_{1}], [χ_{1}, χ_{2}], \dots [χ_{n - 1}, χ_{n}]

is called a partition of $[𝒶, 𝒷]$ .

Riemann sum

Let $f (x)$ be defined on $[𝒶, 𝒷]$ and $𝒫$ be a partition of $[𝒶, 𝒷]$ . Let $Δ χ_{i} = χ_{i} - χ_{i - 1}$ and $𝒸_{i} \in [χ_{i - 1}, χ_{i}]$ . A Riemann sum for $f (x)$ on the interval $[𝒶, 𝒷]$ is

\sum_{κ = 1}^{n} f (𝒸_{i}) Δ χ_{i}

Partition norm

Let $𝒫$ be a partition of the interval $[𝒶, 𝒷]$ . The norm of the partition $| 𝒫 |$ is the width of the largest subinterval. If the subintervals all the same width, the set of points forms a regular partition of the interval.

Approximations

left-end: $𝒸_{i} = χ_{i - 1}$
midpoint: $𝒸_{i} = \frac{χ_{i} - χ_{i - 1}}{2}$
right-end: $𝒸_{i} = χ_{i}$
…: lower, upper, random, …