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Chapter 4: Q. 0 (page 384)
Read the section and make your own summary of the material.
1. Mean Value Theorem
2. Signed and Absolute Area
3. Average value of a function
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Calculate the exact value of each definite integral in Exercises 47–52 by using properties of definite integrals and the formulas in Theorem 4.13.
Fill in each of the blanks:
(a)
(b) is an antiderivative of role="math" localid="1648619282178"
(c) The derivative of is
Suppose f is positive on (−∞, −1] and [2,∞) and negative on the interval [−1, 2]. Write (a) the signed area and (b) the absolute area between the graph of f and the x-axis on [−3, 4] in terms of definite integrals that do not involve absolute values.
Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.
(a) A function f for which the signed area between f and the x-axis on [0, 4] is zero, and a different function g for which the absolute area between g and the x-axis on [0, 4] is zero.
(b) A function f whose signed area on [0, 5] is less than its signed area on [0, 3].
(c) A function f whose average value on [−1, 6] is negative while its average rate of change on the same interval is positive.
Use integration formulas to solve each integral in Exercises 21–62. You may have to use algebra, educated guess and-check, and/or recognize an integrand as the result of a product, quotient, or chain rule calculation. Check each of your answers by differentiating.
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