Review Exam 1 You have 1.25 hours to take the exam. You may have a 3 by 5 in. note card, your graphing calculator, pen/pencil and eraser, but nothing else, not even extra paper. You will have to show all work, some calculator programs can find integrals for you, but you will not be allowed to use that feature on the exam. 1. Find the following indefinite integrals. a. Z b. e6θ sin 7θdθ Z c. Z d. Z tan−1 tdt (ln x)2 dx 3 sin2 (x) cos2 (x)dx e. Use trig substitution √ 7x 1 − x4 dx Z e. Use partial fractions Z 3x3 + 2x2 + 21x + 2 dx (x2 + 1)(x2 + 7) 2. Find the following definite integrals. a. π/3 Z 5 tan5 x sec4 xdx 0 b. 2 Z 1 c. 4y 2 − 5y − 12 dy y(y + 2)(y − 3) √ Z π 5θ3 cos(θ2 )dθ √ π/2 3. Use the Trapezoidal Rule, the Midpoint Rule, and Simpson’s Rule to approximate the given integral with the specified value of n = 8. (Round your answers to six decimal places.) Z 4 √ e4 t sin(4t)dt 0 4. Given: Z 1 9 cos(x2 )dx 0 a. Estimate the errors in the approximations T8 and M8 . (Use the fact that the range of the sine and cosine functions is bounded by 1 to estimate the maximum error. Round your answer to seven decimal places.) b. How large do we have to choose n so that the approximations Tn and Mn to the integral are accurate to within 0.0001? (Use the fact that the range of the sine and cosine functions is bounded by 1 to estimate the maximum error.) 5. Determine whether the following integrals are convergent or divergent. If they are convergent, find their value. a. ∞ Z 1 b. Z 0 √ e− x √ dx x 1 20 dx x5 6. Find the exact area of the surface obtained by rotating the curve about the x-axis. y= 1 1 x3 + , ≤x≤1 6 2x 2 7. A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. (Round your answers to the nearest whole number. Use 9.8m/s2 for the acceleration due to gravity. Recall that the weight density of water is 1000kg/m3 .) Estimate the hydrostatic force on the top of the cube. Answers 1. Find the following indefinite integrals. a. − 6 7 6θ e cos(7θ) + e6θ sin(7θ) + C 85 85 b. t tan−1 (t) − 1 ln | 1 + t2 | +C 2 c. x(ln(x))2 − 2x ln(x) + 2x + C d. 3 3 x− sin(4x) + C 8 32 e. Use trig substitution √ 7 (sin−1 (x2 ) + x2 1 − x4 ) + C 4 e. Partial fraction decomposition: 3x 2 + 2 + 1) (x + 7) √ 3 2 7 x ln(x2 + 1) + tan−1 ( √ ) + C 2 7 7 (x2 2. a. 585 , 8 b. 7 5 ln( 83 ), c. − 52 − 5π 4 3. M8 = 768.142523, T8 = 350.404668, S8 = 712.169876 updated 9/16/14 at 11:30 AM 4. Given: a. | EM |≤ 0.0351563,| ET |≤ 0.0703125, b. For Mn : n ≥ 150, for Tn : n ≥ 213 Your answers may be different depending on your value of K. 5. a. convergent, 2e , b. divergent 6. 263π 256 7. 314 N