Line Integrals Reading Assignment: 6.1, Suggested Problems: 6.1: 1,3,4,7,9,12,14,15, 25, 28; 1) a) Consider the quarter-ellipse, E, in the first quadrant parameterized by x 2cos , 0 / 2 . Find the path integral xy ds . E y sin x2 b) Change the parameterization of this curve to y 1 , 0 x 2 and redo the integral. 4 c) How do the parameterizations from a) and b) compare? How do the results from a) and b) compare? Explain. 2) a) Let r r (), a b, be a curve C given in polar coordinates. Let f(x, y) be a function. Derive the formula for C f ds in terms of . b) Use your formula to find the arc-length of the cardiod r = 1 + cos . 3) Find parameterizations for the following curves: a) The line segment from (1, 2,3) to (2, 4, 1) b) A spring with four coils, axis along the x-axis, radius 2, length along x-axis is 1, spiraling clockwise as x increases. c) A spiral drawn on the cone z = r (choose a convenient one). 4) Let x (t ) (t , t 2 , t 3 ), 0 t 2. Let F(x, y, z) = xyi + yzj + zxk. Evaluate 5) Let the path x (t ) be given by (t, t2, 1), 1 t 3. Evaluate x c 2 F ds . dx xy dy xyz dz . 6) Let C be the directed line segment from (2, 1, 3) to (3, 0, 1). Evaluate C x2 dx xy dy dz .