Finite Sums =)

1. 5.1 Estimating with Finite Sums
Greenfield Village, Michigan
Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington

2. 3
If the velocity is not constant,
we might guess that the
2
distance traveled is still equal
to the area under the curve.
1
1 2
Example: V t 1 0 1 2 3 4
8 1 1
1
1 1
1 2
8 8
2
t v
We could estimate the area under the curve by 0 1
drawing rectangles touching at the left corners. 1
1 1
8
This is called the Left-hand Rectangular 1
2
Approximation Method (LRAM). 1
2
1 1 1 3 1
Approximate area: 3
11 1 2 5 5.75 2
8 2 8 4 8

3. 1 3
2
V t 1
8
2
1
0 1 2 3 4
1 1
1
1
1 2 3
8 8
2
We could also use a Right-hand Rectangular Approximation
Method (RRAM).
1 1 1 3
Approximate area: 1 1 2 3 7 7 .7 5
8 2 8 4

4. 1 3
2
V t 1
t 8
v
0 .5 1 .0 3 1 2 5 2
1 .2 8 1 2 5
1 .5
1
1 .7 8 1 2 5
2 .5
2 .5 3 1 2 5
3 .5
0 1 2 3 4
1 .0 3 1 2 5 1 .2 8 1 2 5 1 .7 8 1 2 5 2 .5 3 1 2 5
Another approach would be to use rectangles that touch at
the midpoint. This is the Midpoint Rectangular
Approximation Method (MRAM).
In this example there are four
subintervals.
Approximate area:
As the number of subintervals
6 .6 2 5
increases, so does the accuracy.

5. 1 3
2
V t 1
With 8 subintervals: 8
2
t v
0 .2 5 1 .0 0 7 8 1
1
0 .7 5 1 .0 7 0 3 1
1 .2 5 1 .1 9 5 3 1
0 1 2 3 4
1.38281
1 .7 5
2 .2 5 1.63281
Approximate area:
2 .7 5 1.94531
6 .6 5 6 2 4
3 .2 5 2 .3 2 0 3 1
3 .7 5 2 .7 5 7 8 1
1 3 .3 1 2 4 8 0.5 6 .6 5 6 2 4
width of subinterval

6. Rectangular Approximation
Methods
◦ LRAM: smaller than true area
◦ RRAM: larger than true area
◦ MRAM: closest to true area
More Accurate = More intervals

7. With the trapezoidal rule you take the

LRAM and add the RRAM and then divide
by two.
So we take LRAM (5.75) from the first

problem and the RRAM (7.75) and we add
them to together.
The answer is 13.5 and then we divide

that by two to get our answer:
The answer is 6.75.
