1. given a curve: C: r(t) = (t^2 - 1)i + 2tj, -1<=t<=1 and the
density function p(x,y) = 15*sqrt(x+2). Let p(x,y) be the linear
density of function of a wire modeled by a smooth curve C in the
xy-plane. The mass M of wire is given by. M = Integral of
p(x,y)ds the monment of mass about x and y-axis are defined
respectively as Mx = Integral of yp(x,y)ds My = Integral of
xp(x,y)ds Finally, the center of mass of the wire has
coordinates (average x, average y) average x = My/M , average y
= Mx/M (note: average x: x-bar, average y: y-bar) a) find the
mass of the wire. b) find the coordinates of the center of mass
of the wire c) sketch the wire in xy-plane (sketch curve C) 2.
consider region bounded by the y-axis and one arc of the
circloid with parametric equations. x = 1- cos(theta), y = theta
- sin(theta), 0<= theta<=2pi use a line integral to find the
area of this region. (hint, you might use the integration tables
in the back of the book (simply use integration by part) 3. use
Green's theorem to find work done