1. Integral Defined Functions Day 2 & Day 3
AP Calculus AB

2. Example 1a The graph given is 𝑓 𝑡 =2 on [0,𝑥].
Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡=2𝑥 for different values of 𝑥.
𝐴 𝑥 = 0 2𝑥 2 𝑑𝑡
Using your answer from part (A), fill in the following table 𝒙 −𝟐
(−𝟒) −𝟏
(−𝟐) 𝟎
(𝟎) 𝟏
(𝟐) 𝟐
(𝟒) 𝟑
(𝟔)
𝐴(𝑥) −8 −4 4 8 12 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥)

3. Example 1a: Continued 𝐴 𝑥 = 0 2𝑥 2 𝑑𝑡𝒙 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥) −8 −4 4 8 12
𝐴 𝑥 = 0 2𝑥 2 𝑑𝑡
Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥)
𝐴 𝑥 =4𝑥

4. Complete the Table: Follow Ex 1a Parts (A)-(C)
Examining an Upper Bound of 2𝑥
𝑓(𝑡)
Equation, Interval
𝐴(𝑥)
Integral Defined Function
Equation
𝐴′(𝑥)
𝑓 𝑡 =2
[−2, 2𝑥]
𝐴 𝑥 = −2 2𝑥 2 𝑑𝑡
𝐴 𝑥 =4𝑥+4
𝐴 ′ 𝑥 =4
0, 2𝑥
𝐴 𝑥 = 0 2𝑥 2 𝑑𝑡
𝐴 𝑥 =4𝑥
[2, 2𝑥]
𝐴 𝑥 = 2 2𝑥 2 𝑑𝑡
𝐴 𝑥 =4𝑥−4

5. Example 1b The graph given is 𝑓 𝑡 =2 on [0,𝑥].
Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡= 𝑥 2 for different values of 𝑥.
𝐴 𝑥 = 0 𝑥 2 2 𝑑𝑡
Using your answer from part (A), fill in the following table 𝒙 −𝟐
(𝟒) −𝟏
(𝟏) 𝟎
(𝟎) 𝟏 𝟐 𝟑
(𝟗)
𝐴(𝑥) 8 2 18 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥)

6. Example 1b: Continued 𝐴 𝑥 = 0 𝑥 2 2 𝑑𝑡𝒙 −𝟐
(𝟒) −𝟏
(𝟏) 𝟎
(𝟎) 𝟏 𝟐 𝟑
(𝟗)
𝐴(𝑥) 8 2 18
𝐴 𝑥 = 0 𝑥 2 2 𝑑𝑡
Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥)
𝐴 𝑥 =2 𝑥 2

7. Complete the Table: Follow Ex 1b Parts (A)-(C)
Examining an Upper Bound of 𝑥 2
𝑓(𝑡)
Equation, Interval
𝐴(𝑥)
Integral Defined Function
Equation
𝐴′(𝑥)
𝑓 𝑡 =2
[−1, 𝑥 2 ]
𝐴 𝑥 = −1 𝑥 𝑑𝑡
𝐴 𝑥 =2 𝑥 2 +2
𝐴 ′ 𝑥 =4𝑥
0, 𝑥 2
𝐴 𝑥 = 0 𝑥 2 2 𝑑𝑡
𝐴 𝑥 =2 𝑥 2
[2, 𝑥 2 ]
𝐴 𝑥 = 2 𝑥 2 2 𝑑𝑡
𝐴 𝑥 =2 𝑥 2 −4

8. Example 1c The graph given is 𝑓 𝑡 =2 on [0,𝑥].
Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡=𝑥+1 for different values of 𝑥.
𝐴 𝑥 = 0 𝑥+1 2 𝑑𝑡
Using your answer from part (A), fill in the following table 𝒙 −𝟐
(−𝟏) −𝟏
(𝟎) 𝟎
(𝟏) 𝟏
(𝟐) 𝟐
(𝟑) 𝟑
(𝟒)
𝐴(𝑥) −2 2 4 6 8 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥)

9. Example 1c: Continued 𝐴 𝑥 = 0 𝑥+1 2 𝑑𝑡𝒙 −𝟐
(−𝟏) −𝟏
(𝟎) 𝟎
(𝟏) 𝟏
(𝟐) 𝟐
(𝟑) 𝟑
(𝟒)
𝐴(𝑥) −2 2 4 6 8
𝐴 𝑥 = 0 𝑥+1 2 𝑑𝑡
Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥)
𝐴 𝑥 =2𝑥+2

10. Complete the Table: Follow Ex 1c Parts (A)-(C)
Examining an Upper Bound of 𝑥+1
𝑓(𝑡)
Equation, Interval
𝐴(𝑥)
Integral Defined Function
Equation
𝐴′(𝑥)
𝑓 𝑡 =2
[−1,𝑥+1]
𝐴 𝑥 = −1 𝑥+1 2 𝑑𝑡
𝐴 𝑥 =2𝑥
𝐴 ′ 𝑥 =2
0, 𝑥+1
𝐴 𝑥 = 0 𝑥+1 2 𝑑𝑡
𝐴 𝑥 =2𝑥+2
[2,𝑥+1]
𝐴 𝑥 = 2 𝑥+1 2 𝑑𝑡
𝐴 𝑥 =2𝑥−2

11. Example 1d The graph given is 𝑓 𝑡 =2 on [0,𝑥].
Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡= 1 2 𝑥 for different values of 𝑥.
𝐴 𝑥 = 𝑥 2 𝑑𝑡
Using your answer from part (A), fill in the following table 𝒙 −𝟐
(−𝟏) −𝟏
− 𝟏 𝟐 𝟎
(𝟎) 𝟏
𝟏 𝟐 𝟐
(𝟏) 𝟑
𝟑 𝟐
𝐴(𝑥) −2 −1 1 2 3 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥)

12. Example 1d: Continued 𝐴 𝑥 = 0 1 2 𝑥 2 𝑑𝑡
𝐴 𝑥 = 𝑥 2 𝑑𝑡
Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥)
𝐴 𝑥 =𝑥 𝒙 −𝟐
(−𝟏) −𝟏
− 𝟏 𝟐 𝟎
(𝟎) 𝟏
𝟏 𝟐 𝟐
(𝟏) 𝟑
𝟑 𝟐
𝐴(𝑥) −2 −1 1 2 3

13. Complete the Table: Follow Ex 1d Parts (A)-(C)
Examining an Upper Bound of 1 2 𝑥
𝑓(𝑡)
Equation, Interval
𝐴(𝑥)
Integral Defined Function
Equation
𝐴′(𝑥)
𝑓 𝑡 =2
[−1, 1 2 𝑥]
𝐴 𝑥 = − 𝑥 2 𝑑𝑡
𝐴 𝑥 =𝑥+2
𝐴 ′ 𝑥 =1
0, 1 2 𝑥
𝐴 𝑥 = 𝑥 2 𝑑𝑡
𝐴 𝑥 =𝑥
[2, 1 2 𝑥]
𝐴 𝑥 = 𝑥 2 𝑑𝑡
𝐴 𝑥 =𝑥−4

14. Example 1e The graph given is 𝑓 𝑡 =2 on [0,𝑥].
Write an integral defined function to represent the area under the curve 𝑡=0 to 𝑡=3𝑥−1 for different values of 𝑥.
𝐴 𝑥 = 0 3𝑥−1 2 𝑑𝑡
Using your answer from part (A), fill in the following table 𝒙 −𝟐
(−𝟕) −𝟏
(−𝟒) 𝟎
(−𝟏) 𝟏
(𝟐) 𝟐
(𝟓) 𝟑
(𝟖)
𝐴(𝑥)
−14 −8 −2 4 10 16 𝒙 −𝟏 −𝟐 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥)

15. Example 1e: Continued 𝐴 𝑥 = 0 3𝑥−1 2 𝑑𝑡𝒙 −𝟐
(−𝟕) −𝟏
(−𝟒) 𝟎
(−𝟏) 𝟏
(𝟐) 𝟐
(𝟓) 𝟑
(𝟖)
𝐴(𝑥)
−14 −8 −2 4 10 16
𝐴 𝑥 = 0 3𝑥−1 2 𝑑𝑡
Plot the points of 𝐴 𝑥 , sketch the graph of 𝐴 𝑥 , and write an explicit equation for 𝐴(𝑥)
𝐴 𝑥 =6𝑥−2

16. Complete the Table: Follow Ex 1e Parts (A)-(C)
Examining an Upper Bound of 3𝑥−1
𝑓(𝑡)
Equation, Interval
𝐴(𝑥)
Integral Defined Function
Equation
𝐴′(𝑥)
𝑓 𝑡 =2
[−1, 3𝑥−1]
𝐴 𝑥 = −1 3𝑥−1 2 𝑑𝑡
𝐴 𝑥 =6𝑥
𝐴 ′ 𝑥 =6
0,3𝑥−1
𝐴 𝑥 = 0 3𝑥−1 2 𝑑𝑡
𝐴 𝑥 =6𝑥−2
[2,3𝑥−1]
𝐴 𝑥 = 2 3𝑥−1 2 𝑑𝑡
𝐴 𝑥 =6𝑥−6

17. Table Analysis
In your groups, answer the following questions based upon the data collected in the table.
Find a pattern to determine the explicit equation of 𝐴(𝑥) from the integral defined function representation of 𝐴(𝑥). Hint: How does changing the lower bound affect 𝐴(𝑥)?
Find a pattern to determine 𝐴′(𝑥) from the integral defined function representation of 𝐴 𝑥 . Hint: Consider the upper bound and the integrand.

18. Example 2: Application From AP Central
Let 𝐹 𝑥 = 2 𝑥 𝑓 𝑡 𝑑𝑡 . The graph of 𝑓 on the interval [−2, 6] consists of two line segments and a quarter of a circle, as shown on the right.
Find 𝐹(0) and 𝐹(4)
Determine the interval where 𝐹(𝑥) is increasing. Justify your answer.
Find the x-coordinates of the inflection points of 𝐹(𝑥). Justify your answer.

19. Example 2: Application From AP Central
Find 𝐹(0) and 𝐹(4)
𝐹 0 = 2 0 𝑓 𝑡 𝑑𝑡 =− 0 2 𝑓 𝑡 𝑑𝑡 =−2
𝐹 4 = 2 4 𝑓 𝑡 𝑑𝑡 = 𝜋
Determine the interval where 𝐹(𝑥) is increasing. Justify your answer.
𝐹(𝑥) is increasing on 0, 4 & (4, 6) because 𝐹 ′ 𝑥 =𝑓(𝑥) is positive on those intervals.
Find the x-coordinates of the inflection points of 𝐹(𝑥). Justify your answer.
𝐹(𝑥) has inflection points at 𝑥=2 and 𝑥=4 because 𝑓 ′ 𝑥 = 𝐹 ′′ 𝑥 changes from a positive to negative slope and vice versa at those points.

20. Example 3: Application From AP Central
Let 𝐻 𝑥 = 0 𝑥+2 𝑓 𝑡 𝑑𝑡 , where 𝑓 is defined on the interval [−5, 5] and the graph 𝑓 consists of three line segments, as shown at the right.
Determine the domain of 𝐻(𝑥)
Determine the x-coordinates of and classify the relative extrema of 𝐻(𝑥). Justify your answer.

21. Example 3: Application From AP Central
Determine the domain of 𝐻(𝑥)
Domain: [−7, 3] , −5≤𝑥+2≤5
Determine the x-coordinates of and classify the relative extrema of 𝐻(𝑥). Justify your answer.
𝐻 ′ 𝑥 =𝑓 𝑥+2 =0, 𝑥=−2 is a critical #
At 𝑥=−2, 𝐻(𝑥) has a rel. max because 𝐻 ′ 𝑥 =𝑓(𝑥+2) changes from negative to positive at this point.

22. Exit Ticket for Feedback
Let 𝐺 𝑥 = 0 𝑥 𝑓 𝑡 𝑑𝑡 , where 𝑓 is defined on the interval [0, 17] and the graph 𝑓 consists of two semicircles and one line segment. Horizontal tangents are located at 𝑥=2 and 𝑥=8, and a vertical tangent is located at 𝑥=4.
Find 𝐹(2), 𝐹′(2), and 𝐹′′(2).
For what values of 𝑥 does 𝑓(𝑥) have a relative minimum value?
Justify your answer.