Fundamental Theorems of Calculus 6.4. The First (second?) Fundamental Theorem of Calculus If f is continuous on, then the function has a derivative at.

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Presentation transcript:

Fundamental Theorems of Calculus 6.4

The First (second?) Fundamental Theorem of Calculus If f is continuous on, then the function has a derivative at every point in, and

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:

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

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:

Example Applying the Fundamental Theorem

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

Example Variable Lower Limits of Integration

Example The Fundamental Theorem with the Chain Rule

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.

Second (first?) FTOC

How to Find Total Area Analytically