# ASU MAT 266 - mat266review_all_f2018 (6 pages)

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- School:
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- Course:
- Mat 266 - Calculus for Engineers II

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MAT266 Exam Review 5 5 The Substitution Rule 1 Z 6 x 4 5 dx 2 3 Suppose f is an odd function For a 0 find 6 1 Integration by Parts 4 Z 2 z ln z dz 5 Z t2 3 dt t3 9t 1 Ra a Z f x dx e2x sin x dx 6 2 Trigonometric Integrals and Substitutions 6 Z dx p 2 x 9 x2 7 Z dx p 4x2 1 8 6 3 Partial Fractions 10 Z 5x2 20x 6 dx x3 2x2 x 11 Z sin x cos x dx Z 8x3 13x dx x2 2 2 4 3 9 Z 2 p 3 p x2 3 dx x 6 4 Integration with Tables and C A S Z p u2 a2 up 2 du u 2 a2 p a2 ln u u2 2 12 Use the equation a2 C to evaluate the folZ p lowing integral x x4 9 dx Z Z du x u 13 Use the equation u ln 1 e C to evaluate the following integral dx u 1 e 1 e x2 6 5 Approximate Integration Z 14 Use the Trapezoid Rule with n 4 intervals to approximate sin d 0 Z 15 Use Simpson s Rule with n 4 intervals to approximate sin d 0 6 6 Improper Integrals Evaluate each integral below if it converges If it diverges clearly state that it diverges Z 1 Z 1 Z 1 Z 2 ex 1 dx x p dx 16 e dx 17 dx 19 18 2x 3 x 1 x 0 1 1 e 0 1 x 1 MAT266 Exam Review 7 1 Area Between Curves In each of the following find the areas between the given curves 20 y x2 2 y 21 y 2 x x 0 x 1 x2 y x 22 y 3x3 23 x 3 7 2 Volumes x2 10x y x2 2x y2 x y 1 24 Find the volume of the solid whose base is bounded by y 1 whose vertical cross sections are equilateral triangles x 2 y 1 x2 x 0 and 25 Find the volume of the solid generated by rotating the region bounded by y 2 about the line y 1 26 Find the volume of the solid generated by rotating the region bounded by y y 3 about x axis x2 y 1 p 25 x2 27 Find the volume of the solid generated by rotating the region bounded by y x2 1 y 0 x 0 x 1 about y axis 7 3 Volumes by Cylindrical Shells Using the method of cylindrical shells find the volume of the solid generated by rotating the specified region about the specified line 28 Region bounded by y x 29 Region bounded by x e y2 x3 the x axis 0 x 1 about the y axis the y axis 0 y 1 about the x axis 30 Region bounded by y x3 x 1 y 1 x 1 about the line x 2 7 4 Arc Length Find the arc length for each of the following functions over the specified interval 31 y x3 1 12 2 6 2x 32 y 1 3 x2 0 8 33 y ln cos x 0 4 7 6 Applications to Physics and Engineering 34 A force of 750 pounds compresses a spring 3 inches from its natural length of 15 inches Find the work done in compressing the spring an additional 3 inches 35 A tank in the shape of a right circular cone is half full of water The tank is 6 ft across the top and 8 ft high How much work is done in pumping all of the water out over the top edge of the tank 2 MAT266 Exam Review 8 1 Sequences Determine whether the following sequences converge If they converge find the limit n n o1 n 8 1 7 36 37 ln n n 2 9n n 0 8 2 Series Determine the convergence or divergence of the following series If it converges find the sum 1 1 1 n X X X 1 1 2n 1 2 1 38 39 40 n n n 1 2 3 3 3 n 1 n 2 n 0 n 1 n 0 8 4 Other Convergence Tests Determine whether the following series converge Justify your answer 41 1 X n en n 1 42 1 X 1 p 4 n3 n 1 43 1 X 1 n n n2 3 n 2 44 1 X 2n n 1 n3 8 5 Power Series Find the interval and radius of convergence of the following series 1 X x n 45 10 n 0 1 X 1 n x 2 n 46 n 1 2 n 0 47 1 X n x 2 n n 0 8 6 Representing Functions as Power Series 48 Find a power series representation for f x 49 Find a power series representation for g x Z 3 50 Use power series to evaluate dx 1 x7 1 centered at 0 1 x 1 centered at 1 x 2 0 3 MAT266 Exam Review 8 7 Taylor and Maclaurin Series Find a power series representation for the following function centered at a 52 f x x1 a 51 f x 3x a 0 1 Find the second degree Taylor polynomial centered at a 53 f x e x 2 a 0 54 f x tan x a 4 9 1 Parametric Curves Eliminate the parameter to find a Cartesian equation of the curve Then sketch the curve and indicate with an arrow the direction in which the curve is traces as the parameter increases 55 x t2 4t y 2 t 4 t 1 56 x 1 e2t y et for all t 57 x cos y sec 0 2 9 2 Calculus with Parametric Curves 58 Find the equation of the tangent line for the parametric equations x t3 25t y 25t2 at t 5 59 Find dy dx and d2 y dx2 expressed as a function of t for curve given by x t sin t y t 2 t4 cos t 3 60 Find the length of the curve parametrized by x 3t y 2t 0 t 2 61 Find the length of the curve parametrized by x sin cos y sin cos 0 3 4 9 3 Polar Coordinates 62 Convert the Cartesian equation x 7 into a polar equation of the form r f 63 Convert the polar equation r 6 cos into a Cartesian equation 64 Find the equation of the tangent line to the polar equation r sin2 1 at the point where 4 65 Find the y coordinate of the highest point on the graph of r 4 cos 9 4 Areas and Lengths in Polar Coordinates 66 Find the area of the region bounded by r tan and 6 3 67 Find the area of the region that lies inside r 2 cos and outside of r 1 68 Find the area enclosed by the inner loop of the curve r 1 2 sin 0 2 69 Find the arc length of …

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