1. Second Fundamental Theorem of Calculus
Morro Rock, California
Greg Kelly, Hanford High School, Richland, Washington
Adapted by: Jon Bannon , Siena College
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 Second Fundamental Theorem of Calculus
If f is continuous on , then the function
has a derivative at every point in , and

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

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

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

7. Second 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:
Second 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.)