Chapter 4: Integration
Section 4.1: Area by Riemann Sums
Use a left Riemann sum to obtain the area bounded by the graph of fx=6+x−x2 and the x-axis.
From the graph of fx=6+x−x2 in Figure 4.1.1(a), the interval a,b appears to be −2,3, a supposition corroborated by the Context Panel calculation
6+x−x2= factor −x+2⁢x−3
tutor (which can only be launched from the Tools≻Tutors menu). Figure 4.1.1(b) shows this tutor being used for a left sum.
Figure 4.1.1(a) Graph of 6+x−x2
Figure 4.1.1(b) Riemann Sums tutor used to obtain a left Riemann sum with 10 subintervals
Alternatively, use the Riemann sum tool below to explore not only the left sum, but all six sums in Table 4.1.1. Enter fx, the bounds a and b, and a value for n. Choose a sum-type from the drop-down box and click the Display button to obtain the exact and approximate values of the area, and a graph showing the approximating rectangles.
Riemann Sum Tool
fx= Riemann Sum: leftrightupperlowermidpointrandom
a= b= n=
Exact Area =
Approximate Area =
Riemann Sum =
In the left Riemann sum with n=10, h=b−an=3−−210=12, 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−2=0; that's why there are only nine rectangles visible in the figure. The rightmost rectangle has height fx9 because x9 is the left end of the tenth subinterval.
The "left ends" of the ten subintervals are −2+i⋅12,i=0,…,9, corresponding to the index on the summation symbol. The term x2 in fx evaluates to −2+12 i2 and the terms 6+x evaluate to 6+−2+12i=4+12i.
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→∞ 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.
Tools≻Tasks≻Browse: Calculus - Integral≻Integration≻Riemann Sums≻Left
The Left Riemann Sum
Enter the interval a,b:
Enter the value of n:
The left Riemann sum:
Value of the Riemann sum:
Table 4.1.1(a) Left Riemann Sum task template applied to fx=6+x−x2 on the interval −2,3
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
limn→∞⁡ = 1256→at 5 digits20.833
Table 4.1.1(b) shows how this very same calculation can be implemented interactively.
Context Panel: Assign Function
fx=6+x−x2→assign as functionf
Write the ratio defining the stepsize h
Context Panel: Assign to a Name≻h
3−−2n = 5n→assign to a nameh
Expression palette: Limit and Summation templates
Context Panel: Evaluate and Display Inline
limn→∞∑k=0n−1f−2+k h h = 1256
Table 4.1.1(b) Computing the limit of a left Riemann sum interactively
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