To find the volume of the solid of revolution when the region is rotated about the y-axis, we use the formula:

\(V = \pi \int x^2 \, dy\)

Given \(y = x^2 + 1\), we have \(x^2 = y - 1\).

Thus, the integral becomes:

\(V = \pi \int (y - 1) \, dy\)

We integrate with respect to \(y\) from 1 to 5:

\(V = \pi \left[ \frac{1}{2}y^2 - y \right]_1^5\)

Calculating the definite integral:

\(V = \pi \left( \frac{1}{2}(5)^2 - 5 - \left( \frac{1}{2}(1)^2 - 1 \right) \right)\)

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

\(V = \pi \left( 12.5 - 5 - 0.5 + 1 \right)\)

\(V = \pi \times 8\)

\(V = 8\pi\)

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