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5.4 The Fundamental Theorem

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The Fundamental Theorem of Calculus, Part 1 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. 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. The long way: 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.

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

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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.

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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) We already know this! To evaluate an integral, take the anti-derivatives and subtract.

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Example 10 : The technique is a little different for definite integrals. We can find new limits, and then we don’t have to substitute back. new limit We could have substituted back and used the original limits.

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Example 8: Wrong! The limits don’t match! Using the original limits: Leave the limits out until you substitute back. This is usually more work than finding new limits

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Example 9 as a definite integral : Rewrite in form of ∫ u n du No constants needed- just integrate using the power rule.

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Substitution with definite integrals Using a change in limits

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