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Published byEdgar Kelly Modified over 5 years ago

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Fundamental Theorems of Calculus 6.4

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

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First Fundamental Theorem: 1. Derivative of an integral.

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

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

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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:

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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:

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Example Applying the Fundamental Theorem

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

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Example Variable Lower Limits of Integration

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Example The Fundamental Theorem with the Chain Rule

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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.

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Second (first?) FTOC

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How to Find Total Area Analytically

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