Not knowing how you set it up, I can’t really say where you went astray, but it sounds as if you’re either setting it up wrong or, at best, setting it in what seems to me the slightly harder way to set it up. (The actual mechanics of the calculation may be easier if you set it up to use washers, however.)
Given the way that $R$ is described, I’d use shells, one for each value of $x$ from $1$ to $2$. Since you’re revolving $R$ about the $y$-axis, the radius of the shell at $x$ is just $x$, so the circumference is $2\pi x$. The height of the shell is the distance between its top edge and its bottom edge. The top edge is on the semicircle $y=\sqrt{4-x^2}$, and the bottom edge is on the line $y=1$, so the height of the shell at $x$ is $\sqrt{4-x^2}-1$. Finally, since I’m taking vertical shells, the thickness of the shell at $x$ is $dx$. By the standard formula the volume differential is therefore
$dV=2\pi x\left(\sqrt{4-x^2}-1\right)dx\;,\tag{1}$
and the total volume of the solid of revolution is obtained by ‘summing’ (i.e., integrating) $(1)$ over the interval from $x=1$ to $x=2$:
$V=2\pi\int_1^2x\left(\sqrt{4-x^2}-1\right)dx\;.$