MAT266 Exam Review§5.5 The Substitution Rule1.Z6(x 4)5dx2.Zt2 3t3+9t +1dt3. Suppose f is an odd function. For a>0, findRaaf(x) dx.§6.1 Integration by Parts4.Zz2ln zdz 5.Ze2xsin xdx§6.2 Trigonometric Integrals and Substitutions6.Zdxx2p9 x27.Zdxp4x2+18.Zsin4x cos3xdx9.Z2p3px2 3xdx§6.3 Partial Fractions10.Z5x2+20x +6x3+2x2+ xdx 11.Z8x3+13x(x2+2)2dx§6.4 Integration with Tables and C.A.S.12. Use the equationZpu2 a2du =u2pu2 a2a22lnu +pu2 a2+ C to evaluate the fol-lowing integral:Zxpx4 9 dx13. Use the equationZdu1+eu= u ln(1 + eu)+C to evaluate the following integral:Zx1+ex2dx§6.5 Approximate Integration14. Use the Trapezoid Rule with n =4intervalstoapproximateZ⇡0sin ✓d✓.15. Use Simpson’s Rule with n =4intervalstoapproximateZ⇡0sin ✓d✓.§6.6 Improper IntegralsEvaluate each i ntegral below if it converges. If it diverges, clearly state that it diverges.16.Z10exdx 17.Z11ex1+e2xdx18.Z21dxx319.Z101(x +1)pxdx1MAT266 Exam Review§7.1 Area Between CurvesIn each of the following, find the areas between the given curves.20. y = x2+2,y = x, x =0,x =121. y =2 x2, y = x22. y =3x3 x2 10x, y = x2+2x23. x =3 y2, x = y +1§7.2 Volumes24. Find th e volume of the solid whose base is bounded by y =1x2, y = 1+x2, x =0,andwhose vertical cross-sections are equilater al triangles.25. Find the volume of the solid generated by rotating the region bounded by y =2 x2, y =1about the line y =1.26. Find the volume of the solid generated by rotating the region bounded by y =p25 x2,y =3aboutx-axis.27. Find the volume of the solid generated by rot at i n g the region bounded by y = x2+1,y =0,x =0,x =1abouty-axis.§7.3 Volumes by Cylindrical ShellsUsing the method of cyli n d ri c al shells, find the volume of the sol i d generated by rotating the specifi edregion about the specified line.28. Region bounded by y = x x3,thex-axis (0 x 1) about the y-axis.29. Region bounded by x = ey2,they-axis (0 y 1) about the x-axis.30. Region bounded by y = x3+ x +1,y =1,x = 1 about t he line x =2.§7.4 Arc LengthFind the arc length for each of the following funct i o n s over the specified interval.31. y =x36+12x,⇥12, 2⇤32. (y 1)3= x2,[0, 8] 33. y =ln(cosx), [0,⇡4]§7.6 Applications to Physics and Engineering34. A force of 750 pounds compresses a spring 3 inches from its natural length of 15 inches. Findthe work done in compressing the spring an additional 3 inches.35. Atankintheshapeofarightcircularconeishalffullofwater. Thetankis6ftacrossthetop and 8 ft high. How much work is done in pumping all of the water out over the top edgeof the tank?2MAT266 Exam Review§8.1 SequencesDetermine whether the following sequences converge. If they converge, find the limit.36.nnln no1n=237.⇢7n+89n1n=0§8.2 SeriesDetermine the convergence or divergence of the following series. If it converges, find the sum.38.1Xn=0✓12n13n◆39.1Xn=12n+13n140.1Xn=0✓23◆n1(n +1)(n +2)§8.4 Other Convergence TestsDetermine whether the following series converge. Justify your an swer.41.1Xn=1nen42.1Xn=114pn343.1Xn=2(1)nnn2 344.1Xn=12nn3§8.5 Power SeriesFind the interval and radius of convergence of the foll owing series.45.1Xn=0⇣x10⌘n46.1Xn=0(1)n(x 2)n(n +1)247.1Xn=0n!(x 2)n§8.6 Representing Functions as Power Series48. Find a power series representation for f(x)=11+x,centeredat0.49. Find a power series representation for g(x)=1(1+x)2,centeredat0.50. Use power series to evaluateZ31 x7dx.3MAT266 Exam Review§8.7 Taylor and Maclaurin SeriesFind a power series representation for the following function, centered at a.51. f(x)=3x, a =0 52. f(x)=1x, a = 1Find the second-degree Taylor polynomial, centered at a.53. f(x)=ex/2, a =0 54. f(x)=tanx, a = ⇡4§9.1 Parametric CurvesEliminate the parameter to find a Cartesian equation of the curve. Then sketch the curve andindicate with an arrow the direction in which the curve is traces as the parameter increases.55. x = t2+4t, y =2t, 4 t 156. x =1+e2t, y = et,forallt57. x =cos✓, y =sec✓,0 ✓<⇡2§9.2 Calculus with Parametric Curves58. Find t h e equation of the tangent line for the parametric eq u a ti o n s x = t3+25t, y =25t2t4at t =5.59. Finddydxandd2ydx2expressed as a function of t for curve given by x = t +sint, y = t cos t.60. Find the length of the curve parametrized by x =3t2, y =2t3,0 t 261. Find the length of the curve parametrized by x =sin✓ +cos✓, y =sin✓ cos ✓,0 ✓ 3⇡4§9.3 Polar Coordinates62. Convert the Cartesian equation x =7intoapolarequationoftheformr = f (✓)63. Convert the polar equation r =6cos✓ into a Cartesian equation.64. Find the equatio n of the tang ent line to the polar equation r =sin2✓ +1atthepointwhere✓ =⇡4.65. Find the y-coordin ate of the highest point on th e graph of r =4cos✓§9.4 Areas and Lengths in Polar Coordinates66. Find the area of the re gi o n bounded by r =tan✓ and⇡6 ✓ ⇡3.67. Find the area of the re gi o n that lies inside r =2cos✓ and outside of r =1.68. Find the area enclosed by the inner loop of the curve r =1+2sin✓,0 ✓ 2⇡69. Find the arc length of the curve r =3sin✓,0 ✓ 2⇡34MAT266 Exam Review (Solutions)1. (x 4)6+ C2. 13ln|t3+9t +1| + C3. 04.z33ln z z39+ C5.15e2x(2 sin x cos x)+C6. p9 x29x+ C7.12lnp4x2+1+2x+ C8.15sin5x 17sin7x + C9. 1 p36⇡10. 6ln|x|ln |x +1|9x +1+ C11. 4ln(x2+2)+32(x2+2)+ C12.x24px4 9 94lnx2+px4 9+ C13.x22+12ln(1 + ex2+ C14.⇡80+p2+2+p2+0⇡ 1.89615.⇡120+2p2+2+2p2+0⇡ 2.00516. 117.⇡218. The integral diverges.19. ⇡20.17621.9222. 2423.9224.2p3325.16⇡1526.256⇡327.3⇡228.4⇡1529. ⇡1 1e⇡ 1.98630.29⇡1531.331632.127403/2 43/2⇡ 9.07333. ln(p2+1)⇡ 0.88134. 3375 inch-pounds35.18752⇡ ⇡ 2945.2ft-lb36. Diverges.37. Converges to 0.38.1239. 1240. 241. Converges by r at i o test.42. Diverges by p-series test.43. Converges by al t er n at in g series test.44. Diverges by ratio test.45. Radius: 10. Interval: (10, 10)46. Radius: 1. Interval: [1, 3]47. Converges only at x =2.48.1Xn=0(x)n5MAT266 Exam Review (Solutions)49.1Xn=1(1)nnxn150. C +1Xn=03x7n+17n +151.1Xn=0(x ln 3)nn!52. 1Xn=0(x +1)n53. 1 x2+x2854. 1+2x +⇡4 2x +⇡4255. x = y2 8y +124 2 2 424656. y =px 12 4 61257. y
View Full Document