Calculus II Exam 2 July 23, 2020 Name: Show all work for full credit. EXACT NUMERICAL ANSWERS (not calculator-generated answers) ARE REQUIRED. 1. Evaluate the following limits or indicate that they diverge. a. 3 lim+ (1 + 4x) x x→0 x26 − 1 b. lim 13 x→1 x − 1 2. Determine whether the following integral is convergent, or divergent. Z 2 ∞ √ 7x dx x5 + 2 3. Evaluate the following integral or show that they diverge. Z 0 −∞ e5x dx (e5x + 3)2 4. Find the length of the curve 4 y 2 = (x + 4)3 , 0 ≤ x ≤ 2, y > 0. 9 5. Find the surface area obtained by rotating the curve y = x-axis. √ 5x over [0, 1] about √ 6. Find the centroid of the region enclosed by y = − 4 − x2 and the x-axis. 7. A laminar R ∪ T consists of a rectangle R of density ρ and a triangle of density 2ρ as shown. a. Find the mass of R and the center of mass of R. b. Find the mass of T and the center of mass of T . c. Using additivity of moments, find the center of mass of R ∪ T .
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