A strip separated from the axis leaves a hole when rotated. Subtract the area of the inner disk from the area of the outer disk; both radii are non-negative distances.
\(V=\pi\int_a^b(R^2-r^2)\,\mathrm{d}x\) about the horizontal axis, or \(V=\pi\int_c^d(R^2-r^2)\,\mathrm{d}y\) about the vertical axis, with the corresponding limits.
Rotating \(1\leq x\leq3\), \(0\leq y\leq2\) about the vertical axis gives \(V=\pi\int_0^2(3^2-1^2)\,\mathrm{d}y=16\pi\).
Use \(R^2-r^2\), not \((R-r)^2\). If the strip touches or crosses the axis, there may be no hole.
For a circle \((x-d)^2+(y-k)^2=a^2\) with \(d>a>0\), rotation about the vertical axis has \(R=d+\sqrt{a^2-(y-k)^2}\) and \(r=d-\sqrt{a^2-(y-k)^2}\).
\(R^2-r^2=4d\sqrt{a^2-(y-k)^2}\), with vertical limits \(k-a\) and \(k+a\).
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