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Chapter 4: Q. 8 (page 361)
Explain why the formula for the integral of does not
apply when What is the integral of
Integral of does not apply when because integral of is undefined.
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Calculate the exact value of each definite integral in Exercises 47–52 by using properties of definite integrals and the formulas in Theorem 4.13.
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 [−π, π].
(b) The average value of f(x) = cos x on [0, 2π].
(c) The area of the region between the graphs of f(x) =
Suppose f is a function whose average value on
is and whose average rate of change on
the same interval is . Sketch a possible graph for f .
Illustrate the average value and the average rate of change
on your graph of f.
Suppose on [1, 3] and on (−∞, 1] and [3,∞). Write the area of the region between the graphs of f and g on [−2, 5] in terms of definite integrals without using absolute values .
Use integration formulas to solve each integral in Exercises 21–62. 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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