The parametric equations of a curve are
\(x=\cos t+t \sin t, \quad y=\sin t-t \cos t\)

The arc of the curve joining the point where \(t=0\) to the point where \(t=\frac{1}{2} \pi\) is rotated about the \(x\)-axis through one complete revolution. Find the area of the surface generated, leaving your result in terms of \(\pi\).