To find the volume of the solid obtained by rotating the region about the \(y\)-axis, we use the method of cylindrical shells. The formula for the volume is:

\(V = \pi \int_{1}^{5} (y - 1) \, dy\)

First, integrate \(y - 1\):

\(\int (y - 1) \, dy = \frac{y^2}{2} - y\)

Apply the limits from 1 to 5:

\(V = \pi \left[ \left( \frac{25}{2} - 5 \right) - \left( \frac{1}{2} - 1 \right) \right]\)

Calculate the expression:

\(V = \pi \left[ \frac{25}{2} - 5 - \frac{1}{2} + 1 \right]\)

\(V = \pi \left[ \frac{25}{2} - \frac{1}{2} - 4 \right]\)

\(V = \pi \left[ \frac{24}{2} - 4 \right]\)

\(V = \pi \times 8\)

\(V = 8\pi\)

Thus, the volume obtained is \(8\pi\) or approximately 25.1.