Week 7 Homework Problems Math 125 1 Stewart, section 7.8: #1, 3, 7, 9, 13, 19, 28, 29, 31, 64, 69, 70 2 Stewart, section 8.1: #1, 2, 9, 11, 12, 17, 24 (omit calculator part), 25 (omit calculator part) 3 Let k be greater than 1. a) Write a de nite integral for the arclength of y = xk from x = 0 to x = b. Do not try to solve the integral. b) One case when this integral can be easily evaluated is when k = 32. In that case use a substitution to evaluate the integral and nd a formula for the arclength in terms of b. c) Use an inverse trig substitution to nd a formula for the arclength in the case when k = 2. d) Use Simpson?s Rule with 6 sub-intervals to estimate the arclength in the case when k = 3 and b = 1. 4 The formula for the arc length of a curve given parametrically by (x(t);y(t)), for a t b, is L = Z b a p (x0(t))2 + (y0(t))2 dt: A path of a point on the edge of a rolling circle of radius R is a cycloid, given by x(t) = R(t sin t) y(t) = R(1 cos t) where t is the angle the circle has rotated. Find the length of one \arch" of this cycloid, that is, nd the distance traveled by a small stone stuck in the tread of a tire of radius R during one revolution of the rolling tire. PSfrag replacements t = 0 t = 2 cycloid x(t) = R(t sin t) y(t) = R(1 cos t) 0 2 R 5 The rocket in Problem 3 of Week 4 required the following force when the rocket was at a distance of x from the center of the moon: F(x) = R 2P x2 pounds: a) The total amount of work done raising the payload from the surface (an altitude of 0, so x = R) to an altitude of R (x = 2R) is W = Z b a F(x) dx = Z 2R R R2P x2 dx = mile-pounds. b) How much work will be needed to raise the payload from the surface of the moon to the \end of the universe"?
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