I was working through line integral exercises when I became uncertain as to how to solve this one problem with a difficult parameterization.
Problem:
$$\int_C (x^2+y^2)dx + 4xydy$$ where $C$ is $x^2+y^2 = ax$, $y \geq 0$ with $a$ being a constant.
Work:
It is clear that the curve $C$ will produce an ellipse, however to obtain a parameterization the ellipse must be in the form of $x^2/a^2 + y^2/b^2 = 1$. To obtain a $1$ on the RHS of the equation, we must divide by $ax$, however in doing this we mix $x$s and $y$s, and from there I am lost on the parameterization. Once that is obtained, I anticipate the rest of the problem to be be routine.
It is actually a circle. The equation is $(x-\frac a 2)^{2}+y^{2}=\frac {a^{2}} 4$. You can parametrize it by writing $x=\frac {a} 2+\frac {|a|} 2 \cos \theta, y=\frac {|a|} 2 \sin \theta$.