Calculating the average are of an annulus and the unit circle

Question

I'm trying a problem and came across the following problem. Choose a point at random uniformly from $\overline B_1(0)$, let's say $x$. Fix some $t_1,t_2\in\mathbb R^+$, what's the average area of the intersection of the annulus of inner radius $t_1$ and outer radius $t_2$ centered at $x$ with $B_1(0)$. I know that if $f(x,y)$ is the area of said annulus centered at $(x,y)$ then the value I'm looking for is $$\frac1\pi\int_{B_1(0)}f(x,y)\;dA=\frac1\pi\int_{0}^{2\pi}\int_0^1rf(r\cos\theta,r\sin\theta)\;drd\theta$$} but I'm having trouble actually computing $f$. Could you help me? If it makes the calculations easier, in my particular problem I have $t_1=t-\varepsilon$ and $t_2=t+\varepsilon$ for some fixed $t\in(0,1)$ and $\varepsilon\in(0,t)$