Line integral for a circumference

Question

I'm having trouble with this line integral: $\displaystyle \int_{C} 3x \ ds$ where $C$ is the segment of a quarter of a circumference $x^2+y^2=4$ that goes from $(2,0)$ to $(0,2)$. I know I need to use Green's Theorem, I just don't know where or how to use the equation of the circle that I've been given.

Answer 1

Green's theorem applies to line integrals over closed paths. What you describe is not a closed path.

You can solve this with a contour integral.

$\mathbf r = (2\cos \theta, 2\sin\theta)\\ d\mathbf r = (-2\sin\theta,2\cos\theta)$

And your function is a scalar function, so we will need

$\|d\mathbf r\|= 2$

$\int_0^\frac{\pi}{2} 12\cos\theta\ d \theta$