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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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Suppose f is positive on (−∞, −1] and [2,∞) and negative on the interval [−1, 2]. Write (a) the signed area and (b) the absolute area between the graph of f and the x-axis on [−3, 4] in terms of definite integrals that do not involve absolute values.
Prove that in three different ways:
(a) algebraically, by calculating a limit of Riemann sums;
(b) geometrically, by recognizing the region in question as a trapezoid and calculating its area;
(c) with formulas, by using properties and formulas of definite integrals.
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) =
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.
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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