Q. 25 - Calculus

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.

Step-by-Step Solution

Answer

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

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.

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