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 by the force field.
F(x,y) = (e^x - y^3)i + (cosy + x^3)j
on a particle that travels once clockwise around the unit circle. Use polar coordinates to evaluate the double integral that you obtain.
4. Let C be the triangular path from (0,0) to (2,0) to (0,0). Use Green's Theorem to evaluate.
Integral of xydx + y^5 dy.
Big help would be appriciate.
Thanks a lot Perfect Hacker. I am sorry that I mistyped the questions. The question number 5 is just the information for the number 1. I think you realized my question even I mistyped. If yes, please let me know. Thanks a bunch.
Check your question again. Green's theorem does not work for such a path.
If that is truly the question then the answer is zero.
Because,
Theorem: If the line integral along a path from A to B exists then the line integral along the same path from B to A exists and is the negative.
Thus, going from (0,0) to (2,0) and then going back from (2,0) to (0,0) will give zero.