1. Section 4.3 Day 3 – Review
1. Find the calories burned by a walker burning 300 calories per hour between 6:00pm and 7:30pm.
A) Express the desired quantity as a definite
integral.
B) Evaluate the integral without using a
calculator.
2. Use your calculator to calculate the following: Find the area enclosed between the x-axis and the graph of 𝑦= 𝑥 2 𝑒 −𝑥 from 𝑥=−1 to 𝑥=3

2. Section 4.1 DAY 1 Antiderivatives and Indefinite Integration
2/17/2019 7:46 PM
Section 4.1 DAY 1 Antiderivatives and Indefinite Integration
AP Calculus AB
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3. Learning Targets
Define antiderivative and its relationship to the derivative
Define constant of integration & its significance to the antiderivative
Define general solution
Operate with basic integration rules
Determine a particular solution from initial conditions
Solve word problems in a context using antiderivatives

4. 𝐶 is the constant of integration
Definition:
𝐹 is an antiderivative of 𝑓 on an interval, if 𝐹 ′ 𝑥 =𝑓(𝑥) for all 𝑥 in the interval.
Notation:
𝑓(𝑥) 𝑑𝑥=𝐹 𝑥 +𝐶
𝐶 is the constant of integration
Antiderivative
Examples:
Let 𝐹 𝑥 =2𝑥+3 and
𝑓 𝑥 =2. Notice that 𝐹 ′ 𝑥 =𝑓(𝑥). Thus, 𝐹(𝑥) is an antiderivative of 𝑓(𝑥).
Notes:
Constant of integration is necessary when providing a general solution.

5. Antiderivatives betterexplained.com
Here’s a metaphor to consider:
Let a plate be a function
Taking the derivative of the function would be breaking the plate into different pieces.
Taking the integral is weighing the broken pieces to have an idea of how “big” it was. It doesn’t tell us what the plate actually looked like.
Taking the antiderivative is figuring out what the actual original shape of the plate was from the broken pieces

6. Constant of Integration
On your own, take the derivative of the following functions.
1. 𝑓 𝑥 = 𝑥 2 +5𝑥−3
2. 𝑓 𝑥 = 𝑥 2 +5𝑥+10
3. 𝑓 𝑥 = 𝑥 2 +5𝑥+𝑒
In your groups, discuss what you notice about the derivatives.

7. Constant of Integration
Let the derivative be 𝑓 𝑥 =4𝑥. (broken shards)
In your groups, come up with a possible answer for the antiderivative. (shape of the plate)

8. Constant of Integration
“+C” is NECESSARY for general solutions
Not enough information is given to determine what the exact constant is
WARNING! If you do not include “+C” for a general solution, it will be INCORRECT and points will be taken off!

9. Constant of Integration
Derivative: 𝑓 𝑥 =4𝑥
To find general solution: 4𝑥 𝑑𝑥
General Solution: 4𝑥 𝑑𝑥=2 𝑥 2 +𝐶
Specific Solution: 2 𝑥 2 +1, 2 𝑥 2 −3, etc.
To find the specific solution, we need more information.

10. Basic Integration Rules
Let 𝑘 be any constant.
1. 𝑘 𝑑𝑥=𝑘𝑥+𝐶
Example: 2𝑑𝑥 =2𝑥+𝐶
2. 𝑘𝑓 𝑥 𝑑𝑥 =𝑘 𝑓 𝑥 𝑑𝑥
Example: 4 𝑥 2 𝑑𝑥 =4 𝑥 2 𝑑𝑥

11. Basic Integration Rules
3. 𝑓 𝑥 ±𝑔 𝑥 𝑑𝑥 = 𝑓 𝑥 𝑑𝑥 ± 𝑔(𝑥) 𝑑𝑥
Example: 𝑥 2 +𝑥 𝑑𝑥= 𝑥 2 𝑑𝑥 + 𝑥𝑑𝑥
Power Rule for Integrals
4. 𝑥 𝑛 𝑑𝑥 = 𝑥 𝑛+1 𝑛+1 +𝐶, where 𝑛≠−1
Example: 𝑥 5 𝑑𝑥 = 𝑥 𝐶= 1 6 𝑥 6 +𝐶

12. Basic Integration Rules –PG 250
cos 𝑥 𝑑𝑥 = sin 𝑥 +𝐶
sin 𝑥 𝑑𝑥 =− cos 𝑥 +𝐶
sec 2 𝑥 𝑑𝑥 = tan 𝑥 +𝐶
sec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 +𝐶
csc 2 𝑥 𝑑𝑥 =− cot 𝑥 +𝐶
csc 𝑥 cot 𝑥 𝑑𝑥 = −csc 𝑥 +𝐶

13. Example 1 - Introductory
Find 3𝑥 𝑑𝑥
3 𝑥 𝑑𝑥=3 𝑥 𝐶=3 𝑥 𝐶
= 3 2 𝑥 2 +𝐶

14. Example 2 - Introductory
Find 2 sin 𝑥 −4 𝑑𝑥
2 sin 𝑥 𝑑𝑥 − 4 𝑑𝑥 =2 − cos 𝑥 −4𝑥+𝐶
=−2 cos 𝑥 −4𝑥+𝐶

15. Example 3 - Introductory
Find 3 𝑥 4 −5 𝑥 2 +𝑥 𝑑𝑥
3 𝑥 4 𝑑𝑥 −5 𝑥 2 𝑑𝑥+ 𝑥 𝑑𝑥
=3 𝑥 −5 𝑥 𝑥 𝐶
= 3 5 𝑥 5 − 5 3 𝑥 𝑥 2 +𝐶

16. Example 4 – Rewriting Find 1 𝑥 3 𝑑𝑥 First, re-write into 𝑥 −3 𝑑𝑥
𝑥 −3 𝑑𝑥 = 𝑥 −3+1 −3+1 +𝐶= 𝑥 −2 −2 +𝐶
=− 1 2 𝑥 2 +𝐶

17. Example 5 - Rewriting Find 𝑥 𝑑𝑥 First, re-write into 𝑥 1 2 𝑑𝑥
𝑥 𝑑𝑥= 𝑥 𝐶= 𝑥 𝐶
= 2 3 𝑥 𝐶

18. Example 6 – Rewriting Find 𝑥+1 𝑥 𝑑𝑥
Re-write into 𝑥+1 𝑥 𝑑𝑥 = 𝑥 𝑥 𝑥 𝑑𝑥
= 𝑥 1− 𝑥 − 𝑑𝑥= 𝑥 𝑥 − 𝑑𝑥
= 𝑥 𝑥 − − 𝐶
= 2 3 𝑥 𝑥 𝐶

19. Exit Ticket for Feedback
𝑥 4 𝑑𝑥
𝑥 𝑥 2 𝑑𝑥