1. This document discusses methods for estimating the area under a curve using finite sums, including left-hand rectangular approximation (LRAM), right-hand rectangular approximation (RRAM), and midpoint rectangular approximation (MRAM). 2. LRAM yields an area that is smaller than the true area, RRAM yields an area that is larger, and MRAM provides the closest approximation to the true area. 3. The document provides an example comparing the different approximation methods and notes that using more subintervals increases the accuracy of the estimates.

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.
