I've been trying to calculate the three-dimensional Lebesgue measure of $\left\{(x,y,\theta)\in\mathbb{R}^2\times[0,\pi):\ x^2+y^2\leq 1;\; \theta\in[0,\pi);\; (x+\cos(\theta))^2+(y+\sin(\theta))^2\leq1\right\}.$
When I was working on it, I tried to do what seemed most natural: using some kind of polar coordinates, which made it seem much more spherical, but I haven't made any further progress on what to make of it because of the last restraint. I tried thinking about this geometrically to get the intuition of where to go with it, but I haven't figured it out yet. Any suggestions?