1. Fundamental Theorem of Calculus
AP CALCULUS AB
Chapter 5:
The Definite Integral
Section 5.4:
Fundamental Theorem of Calculus

2. What you’ll learn about
Fundamental Theorem, Part 1
Graphing the Function
Fundamental Theorem, Part 2
Area Connection
Analyzing Antiderivatives Graphically
… and why
The Fundamental Theorem of Calculus is a Triumph of Mathematical Discovery and the key to solving many problems.

3. Section 5.4 – Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 1
If f is continuous on [a, b], then the function
has a derivative at every point x in [a, b], and

4. The Fundamental Theorem of Calculus

5. Example Applying the Fundamental Theorem

6. Example The Fundamental Theorem with the Chain Rule

7. Example Variable Lower Limits of Integration

8. Section 5.4 – Fundamental Theorem of Calculus
Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of [a, b], and if F is any antiderivative of f on [a, b], then
This is also called the Integral Evaluation Theorem.

9. The Fundamental Theorem of Calculus, Part 2

10. Section 5.4 – Fundamental Theorem of Calculus
Example:

11. Example Evaluating an Integral

12. Section 5.4 – Fundamental Theorem of Calculus
To find the Total Area Analytically
To find the area between the graph of y=f(x) and x-axis over the interval [a, b] analytically:
Partition [a, b] with the zeros of f.
Integrate f over each subdivision.
Add the absolute value of the integrals.

13. Section 5.4 – Fundamental Theorem of Calculus
Ex: Find the area between the curve
and the x-axis over the interval
[-4, 4].

14. Section 5.4 – Fundamental Theorem of Calculus
To find Total Area Numerically (on the calculator)
To find the area between the graph of y=f(x) and the x-axis over the interval
[a, b] numerically, evaluate:
On the TI-89:
nInt (|f(x)|, x, a, b, 50)
On the TI-83 or 84:
fnInt (|f(x)|, x, a, b, 50)

15. Section 5.4 – Fundamental Theorem of Calculus
Application: Average Daily Inventory:
If I(x) is the number of items on hand on day x, the average daily inventory of the items for the period is
If h is the dollar cost of holding one item per day, the average daily holding cost for the period is