Second Fundamental Theorem of Calculus

1. Second Fundamental Theorem of Calculus 5.4

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 2 If f is continuous on , then the function has a derivative at every point in , and

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

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: 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. The Fundamental Theorem of Calculus, Part 1 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