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Quick Review: Fundamental Theorem of Calculus and Analyzing Antiderivatives

Learn about the fundamental theorem of calculus and how it is a key to solving many problems. Explore the derivative of an integral, matching limits of integration, and evaluating integrals analytically. Also, discover how to find total area using these concepts.

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Quick Review: Fundamental Theorem of Calculus and Analyzing Antiderivatives

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  1. Quick Review

  2. Quick Review Solutions

  3. What you’ll learn about • Fundamental Theorem, Part 1 • Graphing the Function • Fundamental Theorem, Part 2 • Area Connection • Analyzing Antiderivatives Graphically … and why The Fundamental Theorem of Calculus is a Triumph of Mathematical Discovery and the key to solving many problems.

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

  5. The Fundamental Theorem of Calculus

  6. 1. Derivative of an integral. First Fundamental Theorem:

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

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

  9. 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.

  10. 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.

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

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

  13. 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.

  14. 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.)

  15. 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. p

  16. The Fundamental Theorem of Calculus, Part 2

  17. Example Evaluating an Integral

  18. How to Find Total Area Analytically Now do example problem 41 on page 303.

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