${}$
Table 4.1.1(a) contains the application of the Left Riemann Sum task template to the function in Example 4.1.1. The path to the task template is given in the heading of the table, and can be accessed either through the Tools menu, or by clicking this link.
Once inside the task template, the Tab key will advance the cursor and ultimately select the first field to be replaced, namely, the field where the function is to be entered. As can be seen in the table, all the fields for this example have been filled in, and the Enter key pressed to execute each command in the table. Note the use of $n$ for the number of subintervals.
The task template implements the RiemannSum command, which accesses the requisite information via equation labels. The option output=sum causes the RiemannSum command to return the left Riemann sum for $n$ subintervals.
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Tools≻Tasks≻Browse: Calculus  Integral≻Integration≻Riemann Sums≻Left

The Left Riemann Sum

Enter $f\left(x\right)$:

${}{{x}}^{{2}}{\+}{x}{\+}{6}$
 (1) 

Enter the interval $\left[a\,b\right]$:

>

$\left[2\,3\right]$

$\left[{}{2}{\,}{3}\right]$
 (2) 

Enter the value of $n$:


The left Riemann sum:

>

$\mathrm{Student}\left[\mathrm{Calculus1}\right]\left[\mathrm{RiemannSum}\right]\left(\,\left[1\right]..\left[2\right]\,\mathrm{method}\=\mathrm{left}\,\mathrm{output}\=\mathrm{sum}\,\mathrm{partition}\=\right)$

$\frac{{5}{}\left({\sum}_{{i}{\=}{0}}^{{n}{}{1}}\phantom{\rule[0.0ex]{5.0px}{0.0ex}}\left({}{\left({}{2}{\+}\frac{{5}{}{i}}{{n}}\right)}^{{2}}{\+}{4}{\+}\frac{{5}{}{i}}{{n}}\right)\right)}{{n}}$
 (4) 

Value of the Riemann sum:

>

$\mathrm{value}\left(\right)$

$\frac{{5}{}\left(\frac{{25}}{{6}}{}{n}{}\frac{{25}}{{6}{}{n}}\right)}{{n}}$
 (5) 




Table 4.1.1(a) Left Riemann Sum task template applied to $f\left(x\right)\=6\+x{x}^{2}$ on the interval $\left[2\,3\right]$



${}$
The Riemann sum ${}$ is evaluated to ${}$ by Maple. In the typical calculus text, the algebra by which this evaluation is effected requires its own chapter to master. Unfortunately, the effort to pass from ${}$ to ${}$ often distracts the student from seeing that the area under a curve is obtained as the limit of a Riemann sum. From the Limit template in the Evaluation palette, with evaluation via the Context Panel, this limit is
$\underset{n\to \infty}{lim}$ = $\frac{{125}}{{6}}$$\stackrel{\text{at 5 digits}}{\to}$${20.833}$${}$
${}$
Table 4.1.1(b) shows how this very same calculation can be implemented interactively.
•

Write $f\left(x\right)\=\dots$
Context Panel: Assign Function


$f\left(x\right)\=6\+x{x}^{2}$$\stackrel{\text{assign as function}}{\to}$${f}$

•

Write the ratio defining the stepsize $h$

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Context Panel: Assign to a Name≻$h$


$\frac{3\left(2\right)}{n}$ = $\frac{{5}}{{n}}$$\stackrel{\text{assign to a name}}{\to}$${h}$${}$

•

Expression palette: Limit and Summation templates

•

Context Panel: Evaluate and Display Inline


$\underset{n\to \infty}{lim}\sum _{k\=0}^{n1}f\left(2\+kh\right)h$ = $\frac{{125}}{{6}}$${}$

Table 4.1.1(b) Computing the limit of a left Riemann sum interactively



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