CALC III Line Integral Problem

  • #1

Homework Statement


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



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.
 
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Answers and Replies

  • #2
vela
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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.

[tex]\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)[/tex]

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.
 

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