# Tutorial: Left Hand Sums

We will work with an example. You can download the Excel File for this example.

The following data is in the given file showing velocity (in feet per second) at time t (given in seconds):

 A B C 1 t v 2 0 1.8 3 0.7 1.2 4 1.4 0.9 5 2.1 0.2 6 2.8 -0.3 7 3.5 -1.0 8 4.2 -0.2 9 4.9 0.7 10 5.6 1.6 11

In this case, we have entered times t given in seconds in column A and velocities v in feet per second in column B. We know that a left hand sum for velocity will approximate distance traveled.

A Left Hand Sum for a given set of data is calculated:
LHS = SUM (delta t first subinterval * v0 + delta t second subinterval * v1 + delta t third subinterval * v2 + delta t fourth subinterval * v3 + . . . + delta t nth subinterval * vn - 1).

Next, we determine the width of each subinterval. We have readings every 0.7 seconds, and as the velocities are in feet/sec, the units agree.
So delta t is 0.7 seconds. (We would have to adjust this if the velocity were given in other units, such as feet/minute or miles/hour.)

We are approximating the Total Change in Distance Traveled between time = 0 and 5.6 seconds with a Left Hand Sum.
Our LHS = delta t *(v(0) + v(0.7) +v(1.4) + . . . + v(4.9))

To do this in Excel:
1. Choose an empty cell below the table,B12, for example.
2. Enter the formula =0.7*SUM(B2:B9).

• An equal sign is needed to tell Excel that you are performing a calculation.
• * is needed to tell Excel to multiply delta t by the sum of the desired numbers of column B.
• SUM(first cell:last cell) will add up all those cells together.
• We use SUM(B2:B9) to add All numbers from B2 through B9 in column B.
3. Hit Return. The result printed in B12 is 2.31, so we conclude that the object moved about 2.31 feet in 5.6 seconds..