how to convert $\int y \, dx$ into the form $\int f(y,dy/ds,s) \,ds$ where $s=\sqrt{x^2+y^2}$

Question

How does one convert $\int y \, dx$ into the form $\int f(y,dy/ds,s) \,ds$ when $s=\sqrt{x^2+y^2}$? I am sure that chain rules would be use to convert the form but I am not sure.

Answer 1

If $s=\sqrt{x^2+y^2}$, then $x=\pm\sqrt{s^2-y^2}$ (sign depend on area of integrating). So

$$dx=d(\pm\sqrt{s^2-y^2})=\frac{\pm (s-y\frac{dy}{ds}) \;ds}{\sqrt{s^2-y^2}}$$

And:

$$\int y dx=\pm \int \frac{\pm y(s-y\frac{dy}{ds}) \;ds}{\sqrt{s^2-y^2}}.$$