1. 6.4 Fundamental Theorem of Calculus
AP Calculus AB
6.4 Fundamental Theorem of Calculus

2. might well be your choice.
Here is a favorite calculus textbook quote, from CALCULUS by Ross L. Finney and George B. Thomas, Jr., ©1990.
If you were being sent to a desert island and could take only one equation with you,
might well be your choice.

3. The Fundamental Theorem of Calculus, Part 1
If f is continuous on , then the function
has a derivative at every point in , and

4. First Fundamental Theorem:
1. Derivative of an integral.

5. First Fundamental Theorem:
1. Derivative of an integral.
2. Derivative matches upper limit of integration.

6. First Fundamental Theorem:
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

7. First Fundamental Theorem:
New variable.
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

8. The long way:
First Fundamental Theorem:
1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

9. 1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

10. 1. Derivative of an integral.
2. Derivative matches upper limit of integration.
3. Lower limit of integration is a constant.

11. The upper limit of integration does not match the derivative, but we could use the chain rule.

12. The upper limit of integration does not match the derivative, but we could use the chain rule.

13. The lower limit of integration is not a constant, but the upper limit is.
We can change the sign of the integral and reverse the limits.

14. Construct a function of the form
That satisfies the given conditions.
Initial x-value
The value of y when x = 2

15. Neither limit of integration is a constant.
We split the integral into two parts.
It does not matter what constant we use!
(Limits are reversed.)
(Chain rule is used.)

16. We already know this! The Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of , and if F is any antiderivative of f on , then
(Also called the Integral Evaluation Theorem)
We already know this!
To evaluate an integral, take the anti-derivatives and subtract.

17. Evaluate the integral.

18. Evaluate the integral.

19. Find the total area of the region between the curve and the x-axis.
Total Area is always positive. + −

20. p