13. LEFT- AND RIGHT- HAND SUMS
The fundamental activity of integral calculus is adding. In the discrete case, we sum a set of values. In the continuous case, we
use the integral to sum over an interval. In this chapter we will restrict our attention to finite discrete sums. You probably already
know that these sums are approximations to the value of a definite integral. We will make that connection in Chapter 16, Riemann
sums.
Distance from the sum of the velocity data
If you drive 50 miles per hour for 3 hours, then you will have traveled 50 + 50 + 50 = 150 miles. If you drive 20 mph for two hours,
then 30 mph for two hours, and finally 40 mph for two hours, then in the six hours you will have traveled 20(2)+ 30(2) + 40(2) =
180 miles. We rarely travel a constant speed. Table 13.1 shows velocity readings at 6 different times. We do not know if we
traveled mostly at 20 ft/sec or 30 ft/sec for the first two seconds. If we assume the velocity is constantly increasing, then these two
numbers give us lower and upper bounds for the first two seconds.
We simply add the first five and multiply by 2, and then repeat this with the last five velocities to find a lower and upper bound on
the distance traveled in ten seconds. We now introduce the technique of summing lists to do this job. The TI has 6 list variables,
named L1,L2,L3,L4,L5,L6. Lists can be created from the home screen using the set symbols {and }then storing (STO > ) the list
in one of the list variable names by pressing 2nd_1 to 2nd_6 .
Creating and summing lists from the home screen:
1
{...}
L
After entering the first data list and storing it in L1(see Figure 13.1), we can use 2nd_ENTRY to edit and create the second list
from the first. Delete the first entry, 20, insert 50 at the end of the list, and change L1to L2. The sum command is under the MATH
submenu of 2nd_LIST .Repeat for the upper sum.
.
Creating and summing lists from STAT EDIT There is an alternate way to enter lists, which we used in Chapters 9 and 10. It is
particularly handy when the list is long and when you need to edit the data. Press STAT 1 :Edit and enter the data in the L1
column. ( Notice that data entered as L1and L2on the home screen will appear in the first two columns. To remove the entire first
column of data, for instance, move the cursor to the heading that highlights L1, pr ess CLEAR ENTER, and the previous data will
be erased.) In Figure 13.2, we repeat the entry and computation above in this setting.
Enter and edit the data in lists by using the numeric and navigation keys.
Paste sum from the MATH submenu of 2nd_List (L1*2) (or from the catalog) and enter
Highlight a list name. Press ENTER and define a list by a formula. Here L1is copied to L2.
The 20 at the top of the list has been deleted and 50 is placed at the end.
We place the sum of L2*2 in L2(6).
