The curve \(C\) has parametric equations
\(x=t^{2}, \quad y=\frac{1}{4} t^{4}-\ln t,\)
for \(1 \leqslant t \leqslant 2\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(y\)-axis.