In the left Riemann sum with $n\=10$, $h\=\frac{b-a}{n}\=\frac{3-\left(-2\right)}{10}\=\frac{1}{2}$, and this is the uniform width of each approximating rectangle. The heights of these rectangles are determined by the values of $f$ at the left end of each rectangle. The leftmost rectangle has height $f\left(-2\right)\=0$; that's why there are only nine rectangles visible in the figure. The rightmost rectangle has height $f\left(x_9\right)$ because $x_9$ is the left end of the tenth subinterval.

The "left ends" of the ten subintervals are $-2\+i\cdot \frac{1}{2}\,i\=0\,\dots \,9$, corresponding to the index on the summation symbol. The term $x^2$ in $f\left(x\right)$ evaluates to ${\left(-2\+\frac{1}{2} i\right)}^2$ and the terms $6\+x$ evaluate to $6\+\left(-2\+\frac{1}{2}i\right)\=4\+\frac{1}{2}i$.

Of course, taking larger values of $n$ will increase the accuracy of the Riemann sum. The next section shows how to obtain a symbolic expression for the Riemann sum with an arbitrary number of subintervals, $n$, in which case the limit as $n\to \infty$ can be calculated. This yields the exact value of the enclosed area.

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.

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.