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Math 152 Class Notes October 29, 2015 Review for Common Exam II 1. Evaluate the following integrals. ˆ (a) dx √ x2 9 − x2 ˆ (b) √ dx x2 − 4x + 13 ˆ (c) x3 √ dx 9x2 − 4 ˆ (d) 3 ˆ (e) 4 x2 dx x−2 x2 + 4x + 4 dx x4 + 4x2 ˆ 2. Evaluate the integral ∞ 2 1 dx or show it diverges. x(ln x)2 ˆ 3. Using the comparison theorem, determine if the integral 1 converges or diverges. ˆ 4. Evaluate the integral 0 3 dx or show it diverges. 4x − 1 ∞ sin2 x + e−x dx x3 5.Find the length of the curve parametrized by x = e3t + e−3t , y = 10 − 6t, 0 ≤ t ≤ 1. 6. A curve is given by x3 = y 2 , 1 ≤ x ≤ 4. Set up both a dx and a dy integral to nd the following surface area. (a) Find the surface area obtained by rotating the curve about the x-axis. (b) Find the surface area obtained by rotating the curve about the y -axis. 2n2 + 1 7. Let an = cos . 3n3 + 4 (a) Determine whether the sequence {an } is convergent. (b) Determine whether the series ∞ P an is convergent. n=1 8. Find the sum of the following series or show it diverges n2 + 1 . (a) The series an , where the sequence of partial sums is sn = 3 n +2 n=1 ∞ P (b) ∞ P sin(nπ). n=1 ∞ 22n P (c) . n n=1 3 9. Let an = ln(n + 1) − ln(n − 1). (a) Determine whether the sequence {an } is convergent. (b) Determine whether the series ∞ P an is convergent. n=2 10. The sequence dended by a1 = Find the limit of the sequence. √ 2, an+1 = √ 2 + an is increasing and bounded.