Chapter 4: Q. 60 (page 327)
Given a simple proof that if n is a positive integer and c is any real number, then
We proved
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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.
Explain why the formula for the integral of does not
apply when What is the integral of
Show by exhibiting a counterexample that, in general, . In other words, find two functions f and g so that the integral of their product is not equal to the product of their integrals.
Consider the region between f and g on [0, 4] as in the
graph next at the left. (a) Draw the rectangles of the left-
sum approximation for the area of this region, with n = 8.
Then (b) express the area of the region with definite
integrals that do not involve absolute values.
Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.
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