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Chapter 4: Q. 68 (page 363)
Solve each of the integrals in Exercises 63鈥�68, where a, b, and c are real numbers with
The value of the integral is
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Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.
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 [鈭�1, 6] is negative while its average rate of change on the same interval is positive.
For each function f and interval [a, b] in Exercises 27鈥�33, use the given approximation method to approximate the signed area between the graph of f and the x-axis on [a, b]. Determine whether each of your approximations is likely to be an over-approximation or an under-approximation of the actual area.
left sum with
a) n = 3 b) n = 6
Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.
(a) True or False: The absolute area between the graph of f and the x-axis on [a, b] is equal to.
(b) True or False: The area of the region between f(x) = x 鈭� 4 and g(x) = on the interval [鈭�3, 3] is negative.
(c) True or False: The signed area between the graph of f on [a, b] is always less than or equal to the absolute area on the same interval.
(d) True or False: The area between any two graphs f and g on an interval [a, b] is given by .
(e) True or False: The average value of the function f(x) = on [2, 6] is
= = 17.(f) True or False: The average value of the function f(x) = on [2, 6] is = = 8.
(g) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 2] and the average value of f on [2, 5].
(h) True or False: The average value of f on [1, 5] is equal to the average of the average value of f on [1, 3] and the average value of f on [3, 5].
Use integration formulas to solve each integral in Exercises 21鈥�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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