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Chapter 4: Q. 5 (page 375)
Use the equation for A(x) that you just found to find A'(x). What do you notice about A'(x) and f (x)?
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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].
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
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 .
Consider the sequence A(1), A(2), A(3),.....,A(n) write our the sequence up to n. What do you notice?
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.
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