1 A curve is given by \(x = t ^ { 2 } - 2 \ln t , y = 4 t\) for \(t > 0\). When the arc of the curve between the points where \(t = 1\) and \(t = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
Given that \(A = k \pi\), where \(k\) is an integer, write down an integral which gives \(A\) and find the value of \(k\).