MATH 152 Exam I - Integration Techniques and Applications - Prof. Z. Nie, Exams of Mathematics

Download MATH 152 Exam I - Integration Techniques and Applications - Prof. Z. Nie and more Exams Mathematics in PDF only on Docsity! MATH 152 Exam I June 11, 2008 1 2 3 4 5 6 7 8 d e b c c a b a For Question 4, I am also giving credit to answer (b). But (c) is the correct answer. The point is that if you use cylindrical shell method, the integral should be ∫ 2 1 2πx ln x dx. Here y = 0 gives the lower limit x = 1. 9. a −1 2 xe−2x − 1 4 e−2x + C b 2 3 x 3 2 ln x − 4 9 x 3 2 + C c 1 2 x − 1 12 sin(6x) + C d −1 3 cos3 x + 1 5 cos5 x + C e 1 4 tan4 x + 1 2 tan2 x + C or 1 4 sec4 x + C since both methods are applicable here. 10. Let x = 2 sin θ. Then dx = 2 cos θdθ. The limits for sin θ = x 2 is from 0 to 1 2 . Therefore ∫ 1 0 √ 4 − x2 dx = ∫ π/6 0 2 cos θ2 cos θ dθ = 4 ∫ π/6 0 cos2 θ dθ =4 ∫ π/6 0 1 2 (1 + cos 2θ) dθ = 2(θ + 1 2 sin 2θ)|π/6 0 = π 3 + √ 3 2 11. 80,000π J