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The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus. Inverse Operations. Fundamental Theorem of Calculus. Discovered independently by Gottfried Liebnitz and Isaac Newton Informally states that differentiation and definite integration are inverse operations. Fundamental Theorem of Calculus.

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The Fundamental Theorem of Calculus

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  1. The Fundamental Theorem of Calculus Inverse Operations

  2. Fundamental Theorem of Calculus • Discovered independently by Gottfried Liebnitz and Isaac Newton • Informally states that differentiation and definite integration are inverse operations.

  3. Fundamental Theorem of Calculus • If a function f is continuous on the closed interval [a, b] and F is an antiderivative of f on the interval [a, b], then

  4. Guidelines for Using the Fundamental Theorem of Calculus • Provided you can find an antiderivative of f, you now have a way to evaluate a definite integral without having to use the limit of a sum. • When applying the Fundamental Theorem of Calculus, the following notation is used

  5. Guidelines It is not necessary to include a constant of integration C in the antiderivative because they cancel out when you subtract.

  6. Evaluating a Definite Integral

  7. Evaluate the Definite Integral

  8. Evaluate the Definite Integral

  9. Definite Integral Involving Absolute Value • Evaluate

  10. Definite Integral Involving Absolute Value

  11. Using the Fundamental Theorem to Find Area • Find the area of the region bounded by the graph of y = 2x3 – 3x + 2, the x-axis, and the vertical lines x = 0 and • x = 2

  12. Using the Fundamental Theorem to Find Area

  13. The Mean Value Theorem for Integrals • If f is continuous on the closed interval [a, b], then there exists a number c in the closed interval [a, b] such that

  14. Average Value of a Function • This is just another way to write the Mean Value Theorem (mean = average in mathematics) • If f is integrable on the closed interval • [a,b], then the average value of f on the interval is

  15. Average Value of a Function

  16. Finding the Average Value of a Function • Find the average value of • f(x) = sin x on the interval [0, p]

  17. Force • The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of sec x, where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is • [0, p/3] and F(0) = 500.

  18. Force • (a) Find F as a function of x. F(x) = 500 sec2 x (b) Find the average force exerted by the press over the interval [0, p/3]

  19. Force

  20. Force

  21. Second Fundamental Theorem of Calculus • If f is continuous on an open interval I containing a, then, for every x in the interval,

  22. Using the Second Fundamental Theorem of Calculus • Evaluate

  23. Second Fundamental Theorem of Calculus • Find F’(x) of

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