Section 8.1 Basic Integration Rules Fitting Integrands to Basic Rules

332460_0801.qxd 11/2/04 3:04 PM Page 524524 CHAPTER 8 Integration Techniques, L’Hôpital’s Rule, and Improper Integrals85. Determine the constants a and b such thatsin x cos x a sinx b.dxUse this result to integratesin x cos x .86. Area The graphs of fx x and gx ax 2 intersect at thepoints 0, 0 and 1a, 1a. Find a a > 0 such that the area2of the region bounded by the graphs of these two functions is 3 .87. Think About It Use a graphing utility to graph the functionfx 1 5x Use the graph to determine whether53 7x 2 10x.fx dx is positive or negative. Explain.088. Think About It When evaluating1 x 2 dx1is it appropriate to substitute u x 2 , x u, and dx to obtain121Explain.Approximation In Exercises 89 and 90, determine which valuebest approximates the area of the region between the x-axis andthe function over the given interval. (Make your selection on thebasis of a sketch of the region and not by integrating.)89. fx 4x 0, 2x 2 1 ,(a) 3 (b) 1 (c) 8 (d) 8 (e) 1090. fx 4 0, 2x 2 1 ,(a) 3 (b) 1 (c) 4 (d) 4 (e) 10Interpreting Integrals In Exercises 91 and 92, (a) sketch theregion whose area is given by the integral, (b) sketch the solidwhose volume is given by the integral if the disk method is used,and (c) sketch the solid whose volume is given by the integral ifthe shell method is used. (There is more than one correct answerfor each part.)201u du 0?2 x 2 dx91. 92.93. Volume The region bounded by y e x 2 , y 0, x 0, andx b b > 0 is revolved about the y-axis.(a) Find the volume of the solid generated if b 1.4(b) Find b such that the volume of the generated solid is 3 cubicunits.94. Arc Length Find the arc length of the graph of y lnsin xfrom x 4 to x 2.95. Surface Area Find the area of the surface formed byrevolving the graph of y 2x on the interval 0, 9 about thex-axis.40 y dydu2u96. Centroid Find the x-coordinate of the centroid of the regionbounded by the graphs of5y y 0, x 0, and x 4.25 x 2,In Exercises 97 and 98, find the average value of the functionover the given interval.97. f x 1 3 ≤ x ≤ 31 x 2,98. f x sin nx, 0 ≤ x ≤ n, n is a positive integer.Arc Length In Exercises 99 and 100, use the integrationcapabilities of a graphing utility to approximate the arc lengthof the curve over the given interval.99. y tan x, 0, 1 4 100. y x 23 , 1, 8101. Finding a Pattern(a) Find cos 3 x dx.(b) Find cos 5 x dx.(c) Find cos 7 x dx.(d) Explain how to find cos 15 x dx without actuallyintegrating.102. Finding a Pattern(a) Write tan 3 x dx in terms of tan x dx. Then find tan 3 x dx.(b) Write tan 5 x dx in terms of tan 3 x dx.(c) Write tan 2k1 x dx, where k is a positive integer, in termsof tan 2k1 x dx.(d) Explain how to find tan 15 x dx without actuallyintegrating.103. Methods of Integration Show that the following results areequivalent.Integration by tables: x 2 1 dx 1 2 xx2 1 ln x x 2 1 CIntegration by computer algebra system: x 2 1 dx 1 2 xx2 1 arcsinhx C104. EvaluatePutnam Exam Challenge42ln9 x dxln9 x lnx 3 .This problem was composed by the Committee on the Putnam Prize Competition.©The Mathematical Association of America. All rights reserved.