1. The Fundamental Theorem of Calculus
AP Calculus
Mrs. Mongold

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

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

6. 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.

7. 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.

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.
We split the integral into two parts.
It does not matter what constant we use!
(Limits are reversed.)
(Chain rule is used.)

12. You already knew this! p The Fundamental Theorem of Calculus
If f is continuous at every point of , and if F is any antiderivative of f on , then
(Also called the Integral Evaluation Theorem)
You already knew this!
To evaluate an integral, take the anti-derivatives and subtract. p