University of Washington Math 324F, Autumn 2014 October 26, 2014 Homework 4 due Tuesday, October 28, 4pm. Problem 1 [Stewart, 16.2.4]. Evaluate the line integral Z x sin y ds, C where C is the line segment from (0, 3) to (4, 6). Problem 2 [Stewart, 16.2.8]. Evaluate the line integral Z x2 dx + y 2 dy, C 2 where C consists of the arc of the circle x + y 2 = 4 from (2, 0) to (0, 2) followed by the line segment from (0, 2) to (4, 3). Problem 3 [Stewart, 16.2.22]. Evaluate the line integral Z xi + yj + xyk · dr, C where C is given by the vector function r(t) = cos ti + sin tj + tk, 0 ≤ t ≤ π. Problem 4 [Stewart, 16.2.36]. Given a curve C with density ρ(x, y, z), we compute its mass and moments Myz , Mxz , and Mxy as follows: Z m= ρ(x, y, z) ds. C The moments about the three coordinate planes are given by Z Z Myz = xρ(x, y, z) ds, Mxz = yρ(x, y, z) ds, C Z Mxy = C zρ(x, y, z) ds. C As before, the center of mass is located at (¯ x, y¯, z¯) = Myz Mxz Mxy , , m m m . Find the mass and center of mass of a wire in the shape of the helix x = t, y = cos t, z = sin t, 0 ≤ t ≤ 2π, if the density at any point is equal to the square of the distance from the origin. Problem 5 [Stewart, 16.2.40]. Find the work done by the force field F(x, y) = x2 i + yex j on a particle that moves along the parabola x = y 2 + 1 from (1, 0) to (2, 1). Problem 6 [Stewart, 16.2.45]. A 160-lb man carries a 25-lb can of paint up a helical staircase that encircles a silo with radius 20 ft. If the silo is 90 ft high and the man makes exactly three complete revolutions climbing to the top, how much work is done by the man against gravity? Answers: 1. − 20 3 cos(6) + 6. 16650 ft·lb. 20 9 sin(6) − 20 9 sin(3). 2. 83 3 . 3. 0. 4. 12π 3 +6π 12 −12π 8π 2 +6 , 8π 2 +6 , 8π 2 +6 . 5. 7 3 + 12 e2 − 21 e.