1. Accumulation AP Calculus AB Days 7-8
Instructional Focus: Define a definite integral and use it to model rate, time, and distance problems. Explain the meaning of a definite integral in the context of the problem.

2. Exploration Accumulation from a Table
Introduce definite integral notation and calculator use.
Introduce Fundamental Theorem of Calculus
Problems 2 and 3 can be assigned as homework then the next day have the students discuss with each other in a group then have group presentations.

3. On a ship at sea, it is easier to measure how fast you are gong than it is to measure how far you have gone. Suppose you are a navigator aboard a supertanker. At 7:30 you are 110 miles from the port you left from. The speed of the ship is measured each 15 minutes and recorded in the table below.
Chunking: Complete problem 1(a)
Have a student read the context of the problem. Then have another summarize the context.

4. Make a plot of the data on the grid below. Be sure to think about units.
Ask the students to plot the data speed (velocity) data. As students work on their plots, circulate and make sure they have appropriate scales and units.
7:30 is not a number, so the horizontal axis should be elapsed time, defined from 7:30 or 7:00. The units of time should be hours. Be sure they have labeled their axes.
Have a student describe how the speed of the ship changed during the 1.75 hour time interval. Sample answer: The speed decreased then increased then decreased and then increased.

5. Make a plot of the data on the grid below. Be sure to think about units.
Ask the students to plot the data speed (velocity) data. As students work on their plots, circulate and make sure they have appropriate scales and units.
7:30 is not a number, so the horizontal axis should be elapsed time, defined from 7:30 or 7:00. The units of time should be hours. Be sure they have labeled their axes.
Have a student describe how the speed of the ship changed during the 1.75 hour time interval. Sample answer: The speed decreased then increased then decreased and then increased.

6. Estimate the distance the ship is from port (the ships position) every 15 minutes from 7:30 p.m. and 9:15 p.m. Explain your answer.
Chunking: Complete Problem 1(b)
Ask the students: If a ship is traveling 28 mph, how far does it go in 15 minutes?
Ask the students to complete part (b). Students may not naturally create a table but do not force them to do a table. After students have completed their work, complete the following table as a class.
We will deal with a consistent notation and way to present the information later as we discuss the results.

7. Estimate the distance the ship is from port (the ships position) every 15 minutes from 7:30 p.m. and 9:15 p.m. Explain your answer.
Time
t hours
x(t) miles
v(t) mph
Chunking: Complete Problem 1(b) Continued
Function notation has been added. Discuss the need for notation that is consistent.
How far did the ship travel during the two hours?
Discuss how the values of x(t) were calculated. Relate this graphically as the sum of the areas of a set of rectangles. Use the graph created in part (a) to draw these rectangles.
Relate the area of each rectangle to the procedure of multiplying 0.25 times v(t).

8. Estimate the distance the ship is from port (the ships position) every 15 minutes from 7:30 p.m. and 9:15 p.m. Explain your answer.
Time
t hours
x(t) miles
v(t) mph
7:30
110 28
7:45
0.25
117 25
8:00
0.50
123.25 20
8:15
0.75
128.25 22
8:30 1
133.75 7
8:45
1.25
135.5 10
9:00
1.5
138 21
9:15
1.75
143.25 26
Chunking: Complete Problem 1(b) Continued
Function notation has been added. Discuss the need for notation that is consistent.
How far did the ship travel during the two hours?
Discuss how the values of x(t) were calculated. Relate this graphically as the sum of the areas of a set of rectangles. Use the graph created in part (a) to draw these rectangles.
Relate the area of each rectangle to the procedure of multiplying 0.25 times v(t).

9. Discussion Do you think the ships speed went from 28 mph to 25 mph in an instance of time? Let’s revisit the graph, if the speed changes continuously, how could we approximate the positions x(t) more accurately.
Chunking: Complete Problem 1(b) Continued
Sample answer: Connect the dots with segments.
Relate this to local linearity.
Have the students redo the table using trapezoids to estimate the distance traveled in a 15-minute time interval. Trapezoidal approximation takes the average of the two velocities to estimate the distance traveled.

10. Discussion Do you think the ships speed went from 28 mph to 25 mph in an instance of time? Let’s revisit the graph, if the speed changes continuously, how could we approximate the positions x(t) more accurately.
Time
t hours
x(t) miles
v(t) mph
7:30
110
26.5
7:45
0.25
22.5
8:00
0.50
122.25 21
8:15
0.75
127.5
14.5
8:30 1
8.5
8:45
1.25
133.25
15.5
9:00
1.5
23.5
9:15
1.75
143
Chunking: Complete Problem 1(b) Continued
Sample answer: Connect the dots with segments.
Relate this to local linearity.
Have the students redo the table using trapezoids to estimate the distance traveled in a 15-minute time interval. Trapezoidal approximation takes the average of the two velocities to estimate the distance traveled.

11. Discussion Why are these values lower than the rectangular sum done previously?
Chunking: Complete Problem 1(b) Continued
If time allows you could get into right and left Reimann sum definition. Related this to a continuous change so the notation changes to 𝑥 𝑡 = 𝑣 𝑡 𝑑𝑡 .
Define this as a definite integral, as the sume of products, graphically the area under a curve. Then move to different forms of this to the fundamental theorem.

12. Discussion What is the relationship between v(t) and x(t)
Discussion What is the relationship between v(t) and x(t)? 𝑣 𝑡 = 𝑥 ′ 𝑡 𝑎 𝑏 𝑥 ′ 𝑡 𝑑𝑡 =𝑥 𝑏 −𝑥 𝑎 𝑥 𝑎 + 𝑎 𝑏 𝑥 ′ 𝑡 𝑑𝑡 =𝑥 𝑏 Definite integral of a rate of change measures the net change.
Chunking: Complete Problem 1(b) Continued
Relate this to other examples besides position and velocity.

13. The velocity of a car, in ft/sec, traveling on
a straight road, for 0 ≤ t ≤ 50 seconds, is
given above in a table of values for v(t).
Estimate with a trapezoidal sum the
distance the car traveled in 50 seconds
using the intervals indicated by the table.
Show how you arrived at your answer.
Emphasize this is the distance traveled, the change in position, not the position. Introduce notation to emphasize the Fundamental Theorem like p(t) is the position of the car at time t, so 𝑣 𝑡 𝑑𝑡 =𝑝 50 −𝑝 0 . Of course position must have a frame of reference and if during the discussion this needs to be addressed in detail, illustrate it does not matter what the value of p(0) is because p(50) – p(0) will not change.

14. In 1993, Kara Hultgreen became one of the
first female pilots authorized to fly navy
planes in combat. Assume that as she
comes in for a landing on the carrier, her
speed in feet per second takes on the values
shown in the table. Find, approximately,
how far her plane travels as it comes to a stop. Is her plane
in danger of running off the other end of the 800-ft-long
flight deck? Justify your answer.
There is no suggestion of whether to use left or right Reimann sums or a trapezoidal sum. Why might you want to use a left Reimann sum? This would give the greatest estimate, the worst case in this context. Better safe than sorry.

15. Exploration 1-3a: Introduction to Definite Integrals
Emphasize area under a curve. This activity uses a graph, not a table of data. Continue to use definite integral notation.

16. As you drive on the highway you accelerate to 100 ft/s to pass a truck
As you drive on the highway you accelerate to 100 ft/s to pass a truck. After you have passed, you slow down to a more moderate 60 ft/s. The diagram shows the graph of your velocity, v(t), as a function of the number of seconds, t, since you started slowing.
Have a student read the context and have another student summarize the context. Have another student describe how the graph is consistent with the context described in words.
Key idea: multiple representations

17. What does your velocity seem to be between t = 30 and
t = 50 s? How far do you travel in the time interval [30, 50]?
Explain why the answer to Problem 1 can be represented as the area of a rectangular region of the graph. Shade this region.
Chunking Problems 1-2
1. 60 ft/sec which means in 20 seconds you would travel (60)(20) = 1200 ft
2. The rectangle is 60 high and has a base of 20 so the area is 1200.
Emphasize how the units become feet and not some square unit.

18. The distance you travel between t = 0 and t = 20 can also be represented as the area of a region bounded by the (curved) graph. Count the number of squares in this region. Estimate the area of parts of squares to the nearest 0.1 square space. For instance, how would you count this partial square?
Chunking Problems 3-5
3. This partial square is about 0.6 and there are about 28.6 squares.

19. How many feet does each small square on the graph represent
How many feet does each small square on the graph represent? How far, therefore, did you go in the time interval [0, 20]?
4. Each square has a base of 5 sec and a height of 10 ft/sec so the area of a small square represents 50 ft. Therefore, the distance is approximately (28.6)(50) = 1430 ft

20. Problems 3 and 4 involve finding the product of the x-value and the y-value for a function where y may vary with x. Such a product is called the definite integral of y with respect to x. Based on the units of t and v(t), explain why the definite integral of v(t) with respect to t in Problem 4 has feet for its units.
5. (seconds)(feet/second) = feet
Emphasize the correct notation for a definite integral here.

21. The graph shows the cross- sectional area, y, in in.2, of a
football as a function of the
distance, x, in in., from one of
its ends. Estimate the definite
integral of y with respect to x.
What are the units of the definite integral in Problem 6? What, therefore, do you suppose the definite integral represents?
Questions 6-8 are optional
6. The square and partial squares under the curve have about 45.2 square spaces of area. Each square space has base representing 1 and height representing 5. So one square space represents 5 units of definite integral, and the total definite integral is about 226 units.
7. The x-units are in inches, and the y-units are in square inches. So their product, the definite integral is in cubic inches. The definite integral seems to represent the volume of the football.