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Chapter 4: Q. 29 (page 404)
The algebra of definite integrals:
Fill in the blanks to complete the definite integral rules that follow. You may assume that f and g are integrable functions on , that , and that k is any real number.
Ans:
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Given a simple proof that if n is a positive integer and c is any real number, then
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.
Verify that(Do not try to solve the integral from scratch.
As n approaches infinity this sequence of partial sums could either converge meaning that the terms eventually approach some finite limit or it could diverge to infinity meaning that the terms eventually grow without bound. which do you think is the case here and why?
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