Question

Express each of the types of Riemann sums that follow in general sigma notation and also as an expanded sum. You may assume that f is a function defined on $[a,b],$n is a positive integer, $∆x=\frac{b-a}{n},\mathrm{and}{\mathrm{k}}_{\mathrm{x}}=\mathrm{a}+\mathrm{k}∆\mathrm{x}.$

lower sum.

Short answer

The general sigma notation of lower sum and its expanded sum is following.

Step-by-step solution

Step 1. Given information.

The given type of Riemann sum is the lower sum.

Step 2. lower sum.

Consider the function defined on the interval $[a,b]$and n is a positive integer.

Then The n-rectangle lower sum for f on $[a,b]$is$\sum _{k=1}^{n}f\left(m_k\right)∆x.$

Where each $m_k$Mk is chosen so that $f\left(m_k\right)$is the minimum value of f on $[x_{\mathrm{k}-1},x_{\mathrm{k}}]$and $∆\mathrm{x}=\frac{\mathrm{b}-\mathrm{a}}{2},{\stackrel{*}{\mathrm{x}}}_{\mathrm{k}}=\mathrm{a}+\mathrm{k}∆\mathrm{x},\&{\stackrel{*}{\mathrm{x}}}_{\mathrm{k}}\in [x_{\mathrm{k}-1},x_{\mathrm{k}}]$

then expanded sum will be the following.

$$\begin{gathered} \sum _{k=1}^{n}f\left(m_k\right)∆x=\sum _{k=1}^{n}f(a+k∆x)∆x \\ \sum _{k=1}^{n}f\left(m_k\right)∆x=\sum _{k=1}^{n}f\left[a+k\left(\frac{b-a}{n}\right)\right]\left(\frac{b-a}{n}\right) \end{gathered}$$