Fundamental Theorem of Calculus

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Presentation transcript:

Fundamental Theorem of Calculus

If you were being sent to a desert island and could take only one equation with you, might well be your choice.

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

First Fundamental Theorem: 1. Derivative of an integral.

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

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

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.

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.

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

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

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.

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.)

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