Question

(1) Length of graphs a) Let a path C be given by the graph of y g(x), a 3 < b, with a piecewise C1 function g : [a, b - IR. Show that the path integral of a continuous function f: IR2- R over the path C is b) Let g : [a, b] - IR be a piecewise C1 function. The length of the graph of g on (t, g(t)). Show that [a,b] is defined as the length of the path [a,Ы-> R2, ț the length of the graph of g on la, is 1 (g(x dr. c) Use part b) to find the length of g: 1,2R, gx) log() (2) Total mass and center of mass of wires a) A thin wire has the shape of the first-quadrant part of the circle with center the origin and radius a 0, The mass density function is δ(x,y-kxy with k > 0. Find the mass and the center of mass of the wire. b) Find the mass and center of mass of a wire in the shape of the helix c(t) - (sin(t), cos(t), t and with mass density function 5(x, y,2)- x2 + y2 + 22 (3) Work done by a force field a) Find the work done by the force field F(a, y)(2,ye") on a particle that b) Find the work done by the force field F(x, y, z) = (z-y,y-z2,2-x2) on a c) Find the work done by the force field F(x,y) - (2x +y, x) on a particle that moves along the parabola -y2+1 from (1,0) to (5,2) particle that moves along the line segment from (1,-2,3) to (2,5,6) moves from (1,1) to (4,3)

Please solve these three questions!

Answer 1

C. 2