It is shown that the most stressful bounded disturbance for linear stable second-order systems with numerator dynamics is bang-bang and can be realized in feedback form using a switch curve. Especially for the second-order systems with a finite zero, an explicit formula is given for the bounded input-bounded output stability integral based on the time maximum disturbance switch curve. This closed analytic form requires less-computational effort and gives an intuitive feeling as to how the L-infinity-gain will change with the free system parameters.