discuss how you would find the area under this curve!
DESCRIPTION
APROXIMATE Area under a Curve AP Calculus Unit 5 Day 2 APROXIMATE Area under a CurveTRANSCRIPT
![Page 1: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/1.jpg)
Discuss how you would find the area under this curve!
![Page 2: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/2.jpg)
AP Calculus Unit 5 Day 2
APROXIMATE Area under a Curve
![Page 3: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/3.jpg)
• We will introduce some new Calculus concepts–LRAM–RRAM–MRAM
![Page 4: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/4.jpg)
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?
2( )f x x
![Page 5: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/5.jpg)
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
2( )f x x
(0)(1)+(1)(1)+(4)(1)=5
![Page 6: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/6.jpg)
Is this an overestimate or an underestimate? What if the function had been a decreasing function?
![Page 7: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/7.jpg)
Is this an overestimate or an underestimate? What if the function had been a decreasing function? Under
SOLUTION
(1)(1)+(4)(1)+(9)(1)=14
![Page 8: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/8.jpg)
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.
![Page 9: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/9.jpg)
Midpoint Method SolutionFind 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.
x 1.5
f 1.5 2 2 21 0.5 1 1.5 1 2.5 8.75
The middle x-values in each region
![Page 10: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/10.jpg)
CAUTION!!!!MRAM ≠ (LRAM + RRAM)/2
![Page 11: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/11.jpg)
Complete the following table:Increasing Function
Decreasing Function
LRAM
RRAM
![Page 12: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/12.jpg)
Under/Over Estimates for Increasing Function
LRAM produces an _____________underestimate RRAM produces an _____________overestimate
![Page 13: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/13.jpg)
LRAM produces an _____________overestimate RRAM produces an _____________underestimate
Under/Over Estimates for Decreasing Function
![Page 14: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/14.jpg)
How do we get the best underestimate of the area between a and b?
a b
Answer: Divide into 2 intervals (increasing/decreasing) and pick LRAM or RRAM for each part that yields an underestimate for that half.
![Page 15: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/15.jpg)
Together let’s practice with finding partition widths
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?
( ) 2f x x 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?
1( )f xx
![Page 16: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/16.jpg)
Area Under Curves
How could we make these approximations more accurate?
You Try:
3 0 3 3 3 6 12.545
3 3 3 6 3 9 21.545
3 1.5 3 4.5 3 7.5 18.254
LRAM
RRAM
MRAM
![Page 17: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/17.jpg)
Let’s look at a HW problem . . .
Unit 5 HW Document
Problem #5
![Page 18: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/18.jpg)
LRAM - Using a table
x 0 3 6 9 12f(x) 20 30 25 40 37
LRAM 3 20
Partitions
Partition [0,3] [3,6] [6,9] [9,12]
3 30 3 25 3 40 345
f(x) value at left end of interval.
Width of interval
![Page 19: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/19.jpg)
RRAM - Using a table
x 0 3 6 9 12f(x) 20 30 25 40 37
RRAM 3 30
Partitions
Partition [0,3] [3,6] [6,9] [9,12]
3 25 3 40 3 37 396
f(x) value at right end of interval.
Width of interval
![Page 20: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/20.jpg)
Midpoint Rectangle Sums (MRAM)
x 0 2 4 6 8 10 12f(x) 20 30 25 40 42 32 37
Partitions
Partition midpoints
MRAM 4 30 4 40 4 32 408
Use the middle function value in each interval.
Ex: Calculate the Midpoint Sum using 3, equal-width partitions.
![Page 21: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/21.jpg)
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 . . . .
![Page 22: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/22.jpg)
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.
Example Problem:
Time (secon
ds
0 2 4 6 8 10
Speed (ft/sec
)
30 36 40 48 54 60
“never decreasing”—
Scatterplot
![Page 23: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/23.jpg)
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.
Example Problem:
1. What is the best lower estimate for the distance the car traveled in the first 2 seconds?
“lower estimate”--_________ since increasing function
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
![Page 24: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/24.jpg)
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.
Example Problem:
Time (sec)
0 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?
![Page 25: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/25.jpg)
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)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
![Page 26: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/26.jpg)
You Try: 4. What is the best upper estimate for the total distance traveled during the first 4 seconds?
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
![Page 27: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/27.jpg)
You Try: 5. Continuing this process, what is the best lower estimate for the total distance traveled in the first 10 seconds?
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
![Page 28: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/28.jpg)
You Try: 6. What is the best upper estimate for the total distance traveled in the first 10 seconds?
Time (sec)
0 2 4 6 8 10
Speed (ft/sec)
30 36 40 48 54 60
![Page 29: Discuss how you would find the area under this curve!](https://reader036.vdocument.in/reader036/viewer/2022062503/5a4d1af57f8b9ab059980cd3/html5/thumbnails/29.jpg)
Riemann SumsThese 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?