Determine Parametric Equation, Shell and Washer Method | Calculus II | MATH 166, Quizzes of Calculus

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Math 166 R Quiz2 Solution 1. (8 pts) Find the arc length of the curve determined by the parametric equation x=, y = t° between two points (0,0) and (4,8). Hint: Find the value of the parameter ¢ correspond to the points first. Solution: The given points correspond to the values t = 0 and ¢ = 2 of the parameter, dx dy 2 ai oy) ae eee eSo Uk di 2 oa (Lyrae =) Vi? + Gi at t 0 2 -{ v4t? + 9t4 dé 0 ey =[ tV4 + 9? dt 0 1 “af ie = 55 (408 — 43) 80V10 — 8) 40 Judu (u = 4+ 92) iL =57¢ 2. (5 pts) Set up(but do not evaluate) the integral for the area of the surface of rev- olution generated by revolving the curve y = 3+ 2x — x?, 0 < x < 3 about the a-axis. Solution: f(x) = 3+ 2x — 2, f'(z) =2-22 3 Azan f (3+ 2a — 2)\/1 + (2 — 2x)? de. 0 3. (12 pts) A region bounded by x = y”, y = 2 and x = 0 is revolved about the z-axis. a. Sketch the region b. find the volume of the solid of revolution by using (i) the shell method and (ii) the washer method. Solution: oo, Yetbeeeer Og TR ae xe 2 4A be ey a Qnyy?dy = onli = 81. oO cares dP Dee i a orfte- bee ii. v-[ a(2? — (zx) jae =x f (4 a)de = n(40 — 52 yp = or