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5.3 Definite Integrals and Antiderivatives, p. 285

Learn about definite integrals, antiderivatives, and rules for working with integrals. Discover how to find average values of functions and apply the Mean Value Theorem. Examples and step-by-step explanations included.

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5.3 Definite Integrals and Antiderivatives, p. 285

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  1. AP Calculus AB/BC 5.3 Definite Integrals and Antiderivatives, p. 285

  2. If the upper and lower limits are equal, then the integral is zero. 2. Reversing the limits changes the sign. 1. Constant multiples can be moved outside. 3. Page 285 gives rules for working with integrals, the most important of which are:

  3. 4. Integrals can be added and subtracted. Reversing the limits changes the sign. 1. If the upper and lower limits are equal, then the integral is zero. 2. Constant multiples can be moved outside. 3.

  4. 5. Intervals can be added (or subtracted.) 4. Integrals can be added and subtracted.

  5. Examples:

  6. The average value of a function is the value that would give the same area if the function was a constant:

  7. Example: Use a calculator that can integrate to find the average value of the function on the interval. At what point(s) in the interval does the function assume its average value? MATH 9:fnInt( So, where does the function reach its average value of -2? -3X2-1,X,0,1) ENTER But, since the interval on the integral was [0, 1], we can’t use the negative value.

  8. Mean Value Theorem (for definite integrals) If f is continuous on then at some point c in , What we just used was the mean value theorem for definite integrals that says that for a continuous function, at some point on the interval the actual value will equal the average value.

  9. Examples: Evaluate the integral using antiderivatives: p

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