CALC III Line Integral Problem

[SolitaryRaf]

Homework Statement

[PLAIN]http://img843.imageshack.us/img843/3995/calc3.jpg

The Attempt at a Solution

Im here asking for some help in direction on how to do these problems so that I can find the solution by myself... I would really really appreciate any help anyone could provide ...

I am not sure how to represent the domain as a function, and if anyone could point me in the right direction, I'm sure I could get the answer.

 

[vela]

First, break the contour up into three pieces, C₁, C₂, and C₃, where C₁ is the line segment from the origin to (1,0), C₂ is the circular arc, and C₃ is the line segment from (1,1) back to the origin.

For each segment of the path, parameterize x and y. For example, for C₃, you could write x(t)=1-t and y(t)=1-t where t goes from 0 to 1. Then dx=-dt and dy=-dt. Now write everything in terms of t.

$$\int_{C_3} (\sin x-6x^2y)dx+(3xy^2-x^3)dy = \int_0^1 [\sin (1-t) - 6(1-t)^2(1-t)](-dt) + [3(1-t)(1-t)^2 - (1-t)^3](-dt)$$

The righthand side is just a run-of-the-mill integral of one variable that you can crank out.

Do this for each segment and add the results together to get your final answer.