A thin, non-uniform rod of length \(L\) lies on a frictionless horizontal table and is pivoted about a fixed, frictionless vertical axis passing through one end at \(x = 0\). The linear mass density of the rod is given by \(\lambda(x) = \lambda_0 \left(\dfrac{x}{L}\right)\), where \(\lambda_0\) is a positive constant and \(x\) is the distance from the pivot. The rod rotates in the horizontal plane with a constant angular speed \(\omega\). What is the total kinetic energy of the rotating rod in terms of \(\lambda_0\), \(L\), and \(\omega\)?