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Chapter 4: Q. 75 (page 375)
Use the Fundamental Theorem of Calculus to give alternative proofs of the integration facts shown in Exercises 72鈥�76. You may assume that all functions here are integrable
Proved that
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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.
Verify that. (Do not try to solve the integral from scratch.)
Without calculating any sums or definite integrals, determine the values of the described quantities. (Hint: Sketch graphs first.)
(a) The signed area between the graph of f(x) = cos x and the x-axis on [鈭捪�, 蟺].
(b) The average value of f(x) = cos x on [0, 2蟺].
(c) The area of the region between the graphs of f(x) =
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.
Calculate the exact value of each definite integral in Exercises 47鈥�52 by using properties of definite integrals and the formulas in Theorem 4.13.
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