Solving a Line Integral: Finding the Value of \int -2y dx + x^2 dy

[jero]

Hello all,

I am trying to solve a line integral:

Find the value of \int -2y dx + x^2 dy over the circle x^2 + y^2 = 9

as you can see, this is a line integral, and I am trying to figure a quick way how it should be solved.
I thought of converting coordinates to (sint,cost) which will end up as a trigonometric function which needs to be integrated, but I believe there is a much easier way which I am not sure about.

Also, I am not sure what is the notation f(y) dx + g(x) dy stands for.
Is it the same as [f(y) - i g(x)] dz?


Thanks for any help.

 

[Shooting Star]

...but I believe there is a much easier way which I am not sure about.

Convert it to a area integral using Green's theorem, which relates a double integral over a region to a line integral over the boundary of the region. The integration becomes very easy.

f(x)dx + g(y)dy is not the same as [f(y) - i g(x)] dz. The latter is a complex function whereas the former is not.