Worksheet 23 11/25/2014 Please work in groups and use the given time to think and try to do the problems. If you cannot solve a question please ask me for a hint or just pass that question. 1. Let f (x, y, z) = x + yz, and let C be the line segment from (0, 0, 0) to (6, 2, 2). (a) Calculate f (c(t)) and ds = kc0 (t)kdt for the parametrization c(t) = (6t, 2t, 2t) for 0 ≤ t ≤ 1. R (b) Evaluate C f (x, y, z)ds. R √ 2. Compute the line integral, C f (x, y) ds, of the scalar function f (x, y) = 1 + 9xy over the curve y = x3 for 0 ≤ x ≤ 5. 3. Find the mass of a wire in the shape of a helix r(t) = 3 cos ti + 3 sin tj + 3tk, 0 ≤ t ≤ 2π with constant density function ρ(x, y, z) = 1. 4. Calculate the total mass of a circular piece of wire of radius 4 cm centered at the origin whose mass density is ρ(x, y) = x2 g/cm. R 5. Evaluate the line integral C y dx − x dy, where C is the parabola y = x2 for 0 ≤ x ≤ 2. R 6. If C is the line segment from (4, 4) to (0, 0), find the value of the line integral C (5y 2 i+5x j)·dr. R 7. Evaluate the line integral C F · dr, where F(x, y, z) = xi − 2yj − 5zk and C is given by the vector function r(t) = hsin t, cos t, ti for 0 ≤ t ≤ 3π/2. R 2 8. Compute C xez where C is the piecewise linear path from (0, 0, 1) to (0, 2, 0) to (1, 1, 1). 1