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5.3 Definite Integrals and Antiderivatives

5.3 Definite Integrals and Antiderivatives. Using the Rules for Definite Integrals. Suppose Find each of the following integrals, if possible. Solution. a. b. c. Not enough information given. You cannot assume that integrating over half the interval would give half the integral.

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5.3 Definite Integrals and Antiderivatives

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  1. 5.3 Definite Integrals and Antiderivatives

  2. Using the Rules for Definite Integrals • Suppose • Find each of the following integrals, if possible.

  3. Solution a. b. c. • Not enough information given. You cannot assume that integrating over half the interval would give half the integral. • Not enough information given. We have no information about the function h outside the interval [-1 , 1] • Not enough information given. Same reason as part e.

  4. Finding Bounds for an Integral • Show that the value of is less than 3/2. • The min f · (b – a) is a lower bound for the value of and that the max f · (b – a) is an upper bound. The max value of on [0 , 1] is . This is less than 3/2.

  5. Average (Mean) value

  6. Applying the Definition • Find the average value of f(x) = 4 - x² on [0,3]. Does f actually take on this value at some point in the given interval?

  7. Mean Value Theorem for Definite Integrals

  8. Derivative with respect to x of the integral of f from a to x is simplify f.

  9. Finding an Integral Using Antiderivatives • Find using the formula • Since F(x) = -cos x is an antiderivative of sin x.

  10. More Practice!!!!! • Homework – Textbook p. 291 # 6 – 36 even.

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