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Chapter 4: Q. 9 (page 404)
Riemann sums: Calculate each of the following Riemann sum
approximations for the definite integral of f on [a, b], using the
given value of n.
The midpoint sum foron.
The midpoint sum is.
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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.
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. (Hint for Exercise 54: ).
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.
Fill in each of the blanks:
(a)
(b) is an antiderivative of role="math" localid="1648619282178"
(c) The derivative of is
Prove Theorem 4.13(c): For any real numbers a and b, Use the proof of Theorem 4.13(a) as a guide.
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