Let $U:\mathbb{R}^2\to \mathbb{R}^2$ be a smooth vector function, $y_0\in\mathbb{R}^2$ and $r>0$. I want to transform the following integral $$ \int_0^r\lambda^2\int_{C(y_0,\lambda)} U(z)\;ds(z)\;d\lambda $$ into an integral over $D(y_0,r)$ where $C(y_0,\lambda)$ denotes the circle of center $y_0$ and radius $\lambda$, and $D(y_0,r)$ is the disk centered at $y_0$ with radius $r$.
I know the Polar Conversion Formula but I dont know how to use it in that example. Thank you.