1. The Fundamental Theorem of Calculus
Section 4.4
Calculus BC AP/Dual, Revised ©2017
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§4.4: The Fundamental Theorem of Calculus

2. §4.4: The Fundamental Theorem of Calculus
Theorems
If a function 𝒇 is continuous on 𝒂, 𝒃 and 𝑭 is an antiderivative of 𝒇 on [𝒂, 𝒃], then 𝒂 𝒃 𝒇 𝒙 𝒅𝒙= 𝑭 𝒃 −𝑭 𝒂
The Integral from 𝒂 to 𝒃 of little 𝒇 of 𝒙 is equal to Big 𝑭 of 𝑩 minus Big 𝑭 of 𝑨 which gives the area underneath the curve.
If little 𝒇 is positive and BIG 𝑭 is the antiderivative
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§4.4: The Fundamental Theorem of Calculus

3. Fundamental Theorem of Calculus
§4.4: The Fundamental Theorem of Calculus

4. §4.4: The Fundamental Theorem of Calculus
Example 1
Solve 𝟏 𝟐 𝒙 𝟐 −𝟑 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

5. §4.4: The Fundamental Theorem of Calculus
Example 2
Solve 𝟎 𝟏 𝒙− 𝒙 𝟒 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

6. §4.4: The Fundamental Theorem of Calculus
Example 2
Solve 𝟎 𝟏 𝒙− 𝒙 𝟒 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

7. §4.4: The Fundamental Theorem of Calculus
Your Turn
Solve 𝟏 𝟒 𝟑 𝒙 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

8. §4.4: The Fundamental Theorem of Calculus
Example 3
Solve 𝟎 𝟐 𝟐𝒙−𝟏 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

9. §4.4: The Fundamental Theorem of Calculus
Example 3
Solve 𝟎 𝟐 𝟐𝒙−𝟏 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

10. §4.4: The Fundamental Theorem of Calculus
Example 4
Solve 𝟏 𝟒 𝟑− 𝒙−𝟑 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

11. §4.4: The Fundamental Theorem of Calculus
Example 4
Solve 𝟏 𝟒 𝟑− 𝒙−𝟑 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

12. §4.4: The Fundamental Theorem of Calculus
Your Turn
Solve 𝟎 𝟑 𝒙 𝟐 −𝟒 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

13. §4.4: The Fundamental Theorem of Calculus
Example 5
Find the area of the region bounded by 𝒚=𝟐 𝒙 𝟐 −𝟑𝒙+𝟐, the 𝒙-axis, 𝒙=𝟎, and 𝒙=𝟐.
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§4.4: The Fundamental Theorem of Calculus

14. §4.4: The Fundamental Theorem of Calculus
Example 6
Solve 𝟎 𝝅/𝟒 𝐬𝐞𝐜 𝟐 𝒙 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

15. §4.4: The Fundamental Theorem of Calculus
Your Turn
Solve 𝟎 𝝅/𝟐 𝐬𝐢𝐧 𝒙+𝟑 𝐜𝐨𝐬 𝒙 𝒅𝒙
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§4.4: The Fundamental Theorem of Calculus

16. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝟏 𝟐 𝟏 𝒙 𝟑 𝒅𝒙 (A) − 𝟕 𝟖 (B) − 𝟑 𝟖 (C) 𝟏𝟓 𝟔𝟒 (D) 𝟑 𝟖
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§4.4: The Fundamental Theorem of Calculus

17. AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝟏 𝟐 𝟏 𝒙 𝟑 𝒅𝒙
Vocabulary
Process and Connections
Answer
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§4.4: The Fundamental Theorem of Calculus

18. §4.4: The Fundamental Theorem of Calculus
Assignment
Page 288
5-21 EOO, odd
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§4.4: The Fundamental Theorem of Calculus