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Chapter 4: Q. 9 (page 384)
Describe an example that illustrates that is not equal to .
on [-1,1]
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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.
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.
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. (Hint for Exercise 54: ).
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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) =
Given a simple proof that if n is a positive integer and c is any real number, then
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