1. Integral Defined Functions Day 1
AP Calculus AB

2. Learning Targets
Define and apply the relationship between the area function and the original function
Solve function values using the integral defined function and the graph.

3. Warm Up Problems The graph of 𝑓(𝑥) is shown to the right.
Use the graph to determine the following.
0 4 𝑓(𝑥) 𝑑𝑥
0 −3 𝑓 𝑥 𝑑𝑥

4. Integral Defined Function
𝐹 𝑥 = 𝑎 𝑥 𝑓 𝑡 𝑑𝑡 𝑎: typically a constant, but can sometimes be a variable 𝑥: typically a linear variable, but could be a polynomial In this class, the use of this function is for the area under the curve. *Note: Notice that the integrand has a different variable. This is to differentiate that the integrand is a different function than the overall function.

5. Example 1 =𝑔(1) =𝑔(−2) =𝑔 4 −𝑔(1) =𝑔 4 −𝑔(−3) 1. 0 1 𝑓 𝑡 𝑑𝑡
Let 𝑓(𝑡) be defined by the graph shown on the right.
Let 𝑔 𝑥 = 0 𝑥 𝑓(𝑡) 𝑑𝑡
A) Rewrite the following integrals in terms of 𝑔.
𝑓 𝑡 𝑑𝑡
2. 0 −2 𝑓 𝑡 𝑑𝑡
𝑓 𝑡 𝑑𝑡
4. −3 4 𝑓 𝑡 𝑑𝑡
=𝑔(1)
=𝑔(−2)
=𝑔 4 −𝑔(1)
=𝑔 4 −𝑔(−3)

6. Example 1: Continued =−6 =5 B) Find the following: 1. 𝑔(−3)
Let 𝑓(𝑡) be defined by the graph shown on the right.
Let 𝑔 𝑥 = 0 𝑥 𝑓(𝑡) 𝑑𝑡
B) Find the following:
1. 𝑔(−3)
2. 𝑔 1 −𝑔(−2)
=−6 =5

7. Example 2 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 for different values of 𝑥.
𝐴 𝑥 = 0 𝑥 2 𝑑𝑡
Using your answer from part (A), fill in the following table 𝒙 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥) −4 −2 2 4 6 𝒙 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥)

8. Example 2: Continued 𝐴 𝑥 = 0 𝑥 2 𝑑𝑡𝒙 −𝟐 −𝟏 𝟎 𝟏 𝟐 𝟑
𝐴(𝑥) −4 −2 2 4 6
𝐴 𝑥 = 0 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 =2𝑥

9. Complete the Table: Follow Ex 2 Parts (A)-(C)
𝑓(𝑡)
Equation & Interval
Sketch of Graph
𝐴(𝑥)
Integral Defined Function
Equation
𝑓 𝑡 =2
[−2, 𝑥]
𝐴 𝑥 = −2 𝑥 2 𝑑𝑡
𝐴 𝑥 =2𝑥+4
[−1, 𝑥]
𝐴 𝑥 = −1 𝑥 2 𝑑𝑡
𝐴 𝑥 =2𝑥+2
[0, 𝑥]
𝐴 𝑥 = 0 𝑥 2 𝑑𝑡
𝐴 𝑥 =2𝑥
[1, 𝑥]
𝐴 𝑥 = 1 𝑥 2 𝑑𝑡
𝐴 𝑥 =2𝑥−2
[2, 𝑥]
𝐴 𝑥 = 2 𝑥 2 𝑑𝑡
𝐴 𝑥 =2𝑥−4
[3, 𝑥]
𝐴 𝑥 = 3 𝑥 2 𝑑𝑡
𝐴 𝑥 =2𝑥−6
𝑓(𝑡)
Equation & Interval
Sketch of Graph
𝐴(𝑥)
Integral Defined Function
Equation
𝑓 𝑡 =2
[−2, 𝑥]
[−1, 𝑥]
[0, 𝑥]
𝐴 𝑥 = 0 𝑥 2 𝑑𝑡
𝐴 𝑥 =2𝑥
[1, 𝑥]
[2, 𝑥]
[3, 𝑥]

10. Table Analysis
In your groups, answer the following questions based upon the data collected in the table.
What is the relationship between the graphs of 𝑓(𝑡) and the graphs of 𝐴(𝑥)?
How does the interval of consideration affect the graph and explicit equation of 𝐴(𝑥)?
What is the relationship between the integral defined function representation of 𝐴(𝑥) and the explicit equation representation of 𝐴(𝑥)?
What is the relationship between 𝐴(𝑥) and 𝑓(𝑡)?

11. Exit Ticket for Feedback
Let 𝑔 𝑥 = 0 𝑥 𝑓 𝑡 𝑑𝑡 . The graph of 𝑓(𝑡) is shown on the right.
A) Find the value of 𝑔 4
B) Find the value of 𝑔 2 −𝑔 −2
C) What function is 𝑔′(𝑥) equivalent
to?