1. 5.4 Fundamental Theorem of Calculus
Morro Rock, California
Greg Kelly, Hanford High School, Richland, Washington
Photo by Vickie Kelly, 1998

2. might well be your choice.
Here is my favorite calculus textbook quote of all time, 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. The upper limit of integration does not match the derivative, but we could use the chain rule.

11. 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.

12. 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.)

13. We already know this! p 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. p