1. Discuss how you would find the area under this curve!

2. APROXIMATE Area under a Curve
AP Calculus Unit 5 Day 2
APROXIMATE Area under a Curve

3. We will introduce some new Calculus concepts
LRAM
RRAM
MRAM

4. Rectangular Approximation Method (RAM) to estimate area under curve
Example Problem: Use LRAM with partition widths of 1 to estimate the area of the region enclosed between the graph of and the x-axis on the interval [0,3].
NOTICE—This is an underestimate. What if the function had been a decreasing function?

5. Rectangular Approximation Method (RAM) to estimate area under curve SOLUTION
Example Problem: Use LRAM with partition widths of 1 to estimate the area of the region enclosed between the graph of and the x-axis on the interval [0,3].
NOTICE—This is an underestimate. What if the function had been a decreasing function? over
(0)(1)+(1)(1)+(4)(1)=5

6. Is this an overestimate or an underestimate
Is this an overestimate or an underestimate? What if the function had been a decreasing function?

7. SOLUTION
(1)(1)+(4)(1)+(9)(1)=14
Is this an overestimate or an underestimate? What if the function had been a decreasing function? Under

8. BEWARE—MRAM uses the average of the two x-values NOT the two y-values!
MRAM gives the best estimate regardless of whether the function is increasing or decreasing.

9. Midpoint Method Solution
Find the area between the x-axis and y = x2 between [0,3] with
partition widths of 1.
MRAM: The Midpoint of each rectangle touches the graph.
The middle x-values in each region

10. CAUTION!!!!
MRAM ≠ (LRAM + RRAM)/2

11. Complete the following table:
Increasing
Function
Decreasing
LRAM
RRAM

12. Under/Over Estimates for Increasing Function
LRAM produces an _____________
underestimate
RRAM produces an _____________
overestimate

13. Under/Over Estimates for Decreasing Function
LRAM produces an _____________
overestimate
RRAM produces an _____________
underestimate

14. How do we get the best underestimate of the area between a and b?
Answer: Divide into 2 intervals (increasing/decreasing) and pick LRAM or RRAM for each part that yields an underestimate for that half.

15. Together let’s practice with finding partition widths
Find the upper and lower approximations of the area under
on the interval [0,2] using 8 partitions. What is the width of each partition?
Find the upper and lower approximations of the area under
on the interval [1,2] using 5 partitions. What is the width of each partition?

16. How could we make these approximations more accurate?
Area Under Curves
You Try:
How could we make these approximations more accurate?

17. Let’s look at a HW problem . . .
Unit 5 HW Document

18. LRAM - Using a table Partitions x 3 6 9 12 f(x) 20 30 25 40 373 6 9 12
f(x) 20 30 25 40 37
Partition [0,3]
[3,6]
[6,9]
[9,12]
Width of interval
f(x) value at left end of interval.

19. RRAM - Using a table Partitions x 3 6 9 12 f(x) 20 30 25 40 373 6 9 12
f(x) 20 30 25 40 37
Partition [0,3]
[3,6]
[6,9]
[9,12]
Width of interval
f(x) value at right end of interval.

20. Midpoint Rectangle Sums (MRAM)
Use the middle function value in each interval.
Ex: Calculate the Midpoint Sum using 3, equal-width partitions.
Partitions x 2 4 6 8 10 12
f(x) 20 30 25 40 42 32 37
Partition midpoints

21. NOTES: Background Problem
A car is traveling at a constant rate of 55 miles per hour from 2pm to 5pm. How far did the car travel?
Numerical Solution—
Graphical Solution—
But of course a car does not usually travel at a constant rate

22. Example Problem:
A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.
Time (seconds 2 4 6 8 10
Speed (ft/sec) 30 36 40 48 54 60
“never decreasing”—
Scatterplot

23. Example Problem:
A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.
Time (sec) 2 4 6 8 10
Speed (ft/sec) 30 36 40 48 54 60
1. What is the best lower estimate for the distance the car traveled in the first 2 seconds?
“lower estimate”--_________ since increasing function

24. Example Problem:
A car is traveling so that its speed is never decreasing during a 10-second interval. The speed at various points in time is listed in the table below.
Time (sec) 2 4 6 8 10
Speed (ft/sec) 30 36 40 48 54 60
2. What is the best upper estimate for the distance the car traveled in the first 2 seconds?

25. You Try:
3. What is the best lower estimate for the total distance traveled during the first 4 seconds? (Assume 2 second intervals since data is in 2 second intervals)
Time (sec) 2 4 6 8 10
Speed (ft/sec) 30 36 40 48 54 60

26. You Try:
4. What is the best upper estimate for the total distance traveled during the first 4 seconds?
Time (sec) 2 4 6 8 10
Speed (ft/sec) 30 36 40 48 54 60

27. You Try:
5. Continuing this process, what is the best lower estimate for the total distance traveled in the first 10 seconds?
Time (sec) 2 4 6 8 10
Speed (ft/sec) 30 36 40 48 54 60

28. You Try:
6. What is the best upper estimate for the total distance traveled in the first 10 seconds?
Time (sec) 2 4 6 8 10
Speed (ft/sec) 30 36 40 48 54 60

29. Riemann Sums
These estimates are sums of products and are known as Riemann Sums.
7. If you choose the lower estimate as your approximation of how far the car traveled, what is the maximum amount your approximation could differ from the actual distance?