Q. 22 - 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, $∆x=\frac{b-a}{n}, \mathrm{and} {\mathrm{k}}_{\mathrm{x}}=\mathrm{a}+\mathrm{k}∆\mathrm{x}.$

Upper sum

Step-by-Step Solution

Answer

The general sigma notation of the Upper sum and its expanded sum is following.
$\sum _{k=1}^{n}f\left(M_k\right)∆x=\sum _{k=1}^{n}f\left[a+k\left(\frac{b-a}{n}\right)\right]\left(\frac{b-a}{n}\right)$

Step 1. Given information.

The given type of Riemann sum is Upper sum.

Step 2. Upper sum.

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

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

Where each $M_k$is chosen so that $f\left(M_k\right)$is the maximum 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.

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