1. Fundamental Theorem of Calculus

2. 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. 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)
To evaluate an integral, take the anti-derivatives and subtract. p