1. Estimating with Finite Sums
This lesson is an introduction to integration. Specifically, we introduce the concept of accumulating changes over an interval.

2. Congratulations! You are now a train dispatcher
Train Dispatchers are the air traffic controllers of the railroads. They control the movement of trains over large track territories
They use computers and radio communications to control the safe movement of trains

3. Job Requirements Communication skills Math Science Attentive to detail
Safety conscious

4. Skills for Success Understanding speed, time and distance
If a train travels at a constant speed of 50 ft/sec for 20 seconds, what distance does it travel?
Speed
Distance = rate x time
50 ft/sec for 20 sec = 1,000 ft
Most students will readily understand the formula but many have never considered the geometric representation. Ask them to identify the figure it represents. Once they identify a rectangle, then ask them how the distance related to the figure. The distance = area
This formula only works when you have a constant rate, or speed, over a period of time. Think about when you drive. Do you always go the same speed?
Time

5. Skills for Success
How do you determine the distance graphically, if the train is now traveling at a constant speed of 35 ft/sec for 20 seconds, from 1:00:00 p.m. to 1:00:20 p.m.?
After 20 seconds you check the speed again and determine the train is now traveling at 40 ft/sec and continues at this speed for 20 seconds. Determine the overall distance.
What is the difference between the rectangles?
Speed
With varying speeds, how would you represent the distance geometrically?
Ask them to think about the figure they used on the previous slide.
They should draw two rectangles.
Using the concept of area, what is the total distance?
35 ft/sec for 20 sec = 700 ft
40 ft/sec for 20 sec = 800 ft
1500 ft
This equals the sum of the area of the 2 rectangles.
What is the difference between the rectangles?
So what is determining the height of each rectangle?
Time

6. Skills for Success Many factors impact a train’s speed Train weight
Train length
Engineer
Track’s curvature
Speed limit
Physical conditions
What do you think the factors are that could impact a train’s speed?
Once students list some, show them this list or add to it.
Talk about the importance in real life of being able to deal with functions that change over time. This should help you emphasize the importance of calculus!

7. A More Realistic Scenario
A train’s speed is measured every 5 seconds, resulting in the following data: Time (sec) Speed (ft/sec) Can you approximate how far the train traveled?
Formula: y = .01x2 + x – 1
What do you notice about these speeds vs the previous train speeds?
What does it mean to approximate?
Using rectangles, approximate the distance
What is your approximation?
Students may use multiple methods. Some may use LRAM, some RRAM. Have them describe how they placed the rectangles to emphasize the difference is the height.
How did you determine the height?
LRAM: ft
RRAM: ft
MRAM: ft
The actual is ft. Does this surprise you?
How else could you place the rectangles to find an approximation?
Determine the new approximation.
How does this compare with the actual and previous approximation?

8. Exploring Three Methods
Rectangular Approximation Methods
Left-hand endpoint (LRAM)
Right-hand endpoint (RRAM)
Midpoint (MRAM)
The height of the rectangle is determined by the method used
Approximations
Over-estimate
Under-estimate
Tell them they have been using Reimann sums, specifically rectangular approximation methods

9. Summary All approximations compute using Δx·∑f(xi)
LRAM: Dx[f(0) + f(1) + f(2) + … + f(n-1)]
MRAM: Dx[f(M1) + f(M2) + … + f(MN)]
RRAM: Dx[f(1) + f(2) + f(3) + … + f(n)]
Δx = B-A N
M is the midpoint
N = the number of intervals
A and B are the range
What happens when you increase the number of rectangles?
What happens when you reduce the width of the rectangles?
Ideally, how many rectangles do you want?

10. Example: How do we measure distance traveled?
A train is moving with increasing speed. We measure the train’s speed every three seconds and obtain the following data.
Time (sec)
Speed (ft/sec)
Have them use LRAM
How far has the train traveled?

11. Left –hand endpoint Rectangular Approximation Method
Example:
Graphically:
This is called a LRAM
Left –hand endpoint Rectangular Approximation Method
Is it an over or under-estimate? Why?
Would LRAM ever be an over-estimate?

12. Right –hand endpoint Rectangular Approximation Method
Example:
Graphically:
This is called a RRAM
Right –hand endpoint Rectangular Approximation Method
Is it an under or over-estimate?
Now have the calculate RRAM

13. Midpoint Rectangular Approximation Method
Example:
This time let’s take the midpoint:
This is called an MRAM
Midpoint Rectangular Approximation Method
Have the calculate MRAM

14. Example: A train’s speeds are measured, yielding the data below:
Compute LRAM and RRAM using 3 rectangles
Time (sec)
Speed (ft/sec)
LRAM:
Which Is an over-estimate and which is an under-estimate?
Is this what you expected based upon your previous example?
Why is LRAM an over-estimate now but an under-estimate before?
RRAM:

15. Over or Under Estimates
If f(x) is decreasing
LRAM is an over-estimate
RRAM is an under-estimate
Rn < area < Ln
If f(x) is increasing
LRAM is an under-estimate
RRAM is an over-estimate
Ln < area < Rn

16. Example: What if we changed the number of intervals?
What is the ideal number of intervals?
What technique have we learned about that helps us determine an infinite number of intervals?

17. RAM
If you increase the number of intervals (rectangles), your approximations become increasingly more accurate
What if we take the limit as the number of intervals→∞?
This should give us the exact area under the curve
If f(x) is continuous on [a,b], then the endpoint and midpoint approximations approach one and the same limit L:
lim RN = lim LN = lim MN = L
N→∞ N→∞ N→∞
If f(x)>0 on [a,b], we take L as the definition of the area under the graph of y = f(x) over [a,b]

18. Example:
Sketch the graph of y = 8 – x between the region and the x-axis 1<x<5 into 4 subintervals.
Compute LRAM, MRAM and RRAM.
n = a = b =
LRAM:
Dx[f(0) + f(1) + f(2) + … + f(n-1)]
RRAM:
Dx[f(1) + f(2) + f(3) + … + f(n)]
MRAM:
Dx[f(m1) + f(m2) + f(m3) +… + f(mn)]

19. Example:
Sketch the graph of y = 2 + 4x – x2 between the region and the x-axis 0<x<4 into 6 subintervals.
Compute LRAM, MRAM and RRAM.
LRAM:
Dx[f(0) + f(1) + f(2) + … + f(n-1)]
RRAM:
Dx[f(1) + f(2) + f(3) + … + f(n)]
MRAM:
Dx[f(m1) + f(m2) + f(m3) +… + f(mn)]

20. Summary We can approximate the area under a curve using rectangles
There are three methods for approximating:
LRAM
MRAM
RRAM
The method will determine the heights of the approximating rectangles
Increasing the number of rectangles makes the approximation more accurate
The increasing/decreasing nature of the curve impacts the accuracy of the estimate

21. Summary Why is finding an area important?
It is a way to describe how the instantaneous changes accumulate over an interval
We call this integral calculus
Other applications
The work it takes to empty a tank of oil
Accumulation of water in a conical tank
The pressure against a dam at any depth