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5.1 Estimating with Finite Sums
Greenfield Village, Michigan
time
velocity
After 4 seconds, the object has gone 12 feet.
Consider an object moving at a constant rate of 3 ft/sec.
Since rate . time = distance:
If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line.
ft3 4 sec 12 ft
sec
3t d
When we find the area under a curve by adding rectangles, the answer is called a Riemann sum.
211
8V t
subinterval
partition
The width of a rectangle is called a subinterval.
The entire interval is called the partition.
Subintervals do not all have to be the same sizebut for today they will.
If the velocity is not constant,we might guess that the distance traveled is still equalto the area under the curve.
(The units work out.)
211
8V t Example:
We could estimate the area under the curve by drawing rectangles touching at their left corners.
This is called the Left-hand Rectangular Approximation Method (LRAM).
11
18
11
2
12
8
t v
10
1 11
8
2 11
2
3 12
8Approximate area: 1 1 1 3
1 1 1 2 5 5.758 2 8 4
We could also use a Right-hand Rectangular Approximation Method (RRAM).
11
8
11
2
12
8
Approximate area: 1 1 1 31 1 2 3 7 7.75
8 2 8 4
3
211
8V t
Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM).
1.031251.28125
1.78125
Approximate area:6.625
2.53125
t v
1.031250.5
1.5 1.28125
2.5 1.78125
3.5 2.53125
In this example there are four subintervals.As the number of subintervals increases, so does the accuracy.
211
8V t
211
8V t
Approximate area:6.65624
t v
1.007810.25
0.75 1.07031
1.25 1.19531
1.382811.75
2.25
2.75
3.25
3.75
1.63281
1.94531
2.32031
2.75781
13.31248 0.5 6.65624
width of subinterval
With 8 subintervals:
The exact answer for thisproblem is .6.6
Circumscribed rectangles are all above the curve:
Inscribed rectangles are all below the curve:
We will be learning how to find the exact area under a curve if we have the equation for the curve. Rectangular approximation methods are still useful for finding the area under a curve if we do not have the equation.
The TI-89 calculator can do these rectangular approximation problems. This is of limited usefulness, since we will learn better methods of finding the area under a curve, but you could use the calculator to check your work.
If you have the calculus tools programinstalled:
Set up the WINDOW screen as follows:
Select Calculus Tools and press Enter
Press APPS
Press F3
Press alpha and then enter: 1/ 8 ^ 2 1x
Make the Lower bound: 0Make the Upper bound: 4Make the Number of intervals: 4
Press Enter
and then 1
Note: We press alpha because the screen starts in alpha lock.
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