1. AP Calculus Mrs. Mongold

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

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

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

5. 2. Derivative matches upper limit of integration. 3. Lower limit of integration is a constant. First Fundamental Theorem:

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

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

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

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

10. 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.

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

12. 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. 