$\mathbf{A}-\mathbf{1}$. Given a $2 \times 3$ rectangle, with vertices at $(0,0),(2,0),(0,3)$, and $(2,3)$. Rotate it $90^{\circ}$ clockwise around the point $(2,0)$, then $90^{\circ}$ clockwise around the point $(5,0)$, then $90^{\circ}$ clockwise around the point $(7,0)$, and finally $90^{\circ}$ clockwise around the point $(10,0)$ (at this point, the side that was originally on the $x$-axis is back on the $x$-axis). The point initially at $(1,1)$ on the rectangle traces out a curve during these rotations. Find the area of the region enclosed by this curve and the $x$-axis.
As shown in the figure, the area in question includes four right triangles of $1 \times 1$ with an area of $\frac{1}{2}$ each, four right triangles of $1 \times 2$ with an area of 1 each, two quarter-circles with an area of $\frac{\pi}{4}(\sqrt{2})^{2}=\frac{\pi}{2}$ each, and two quarter-circles with an area of $\frac{\pi}{4}(\sqrt{5})^{2}=\frac{5}{4} \pi$ each. Therefore, the total area of the region is $\frac{7}{2} \pi+6$.