Projection of random variable to circle

Question

Suppose I have a normal distribution in the plane $X\sim\text{Normal}(\mu, \text{Id}), X\in \mathbb{R}^2$ with $\mu\in\mathbb{R}^2$. If I map $X\to X / \Vert X\Vert$ then this defines a new random variable $Y$ constrained to $S^1$, the circle. I want give a formula for the density of a point $y\in S^1$ under this transformation. Making the natural inclusion of $S^1$ into $\mathbb{R}^2$ (which I will denote $\mathbf{i} : S^1 \to\mathbb{R}^2$) I thought it would be sufficient to compute \begin{align} p(y) &= \int_{\left\{x ~:~ x / \Vert x\Vert = \mathbf{i}(y)\right\}} \text{Normal}(x;\mu,\text{Id}) ~\mathrm{d}x \\ &= \int_0^\infty \text{Normal}(t~\mathbf{i}(y), \mu, \text{Id}) ~\mathrm{d}t. \end{align} Is this correct? How could I verify my answer with simulation?