Strong gravitational lensing by irregular mass distributions, such as galaxy clusters, is generally not well quantified by cross sections of analytic mass models. Computationally expensive ray-tracing methods have so far been necessary for accurate cross-section calculations. We describe a fast, semi-analytic method here which is based on surface integrals over high-magnification regions in the lens plane and demonstrate that it yields reliable cross sections even for complex, asymmetric mass distributions. The method is faster than ray-tracing simulations by factors of $∼30$ and thus suitable for large cosmological simulations, saving large amounts of computing time. We apply this method to a sample of galaxy cluster-sized dark matter haloes with simulated merger trees and show that cluster mergers approximately double the strong-lensing optical depth for lens redshifts $z_\mathrm{l}\gtrsim0.5$ and sources near $z_\mathrm{s} = 2$. We believe that this result hints at one possibility for understanding the recently detected high arcs abundance in clusters at moderate and high redshifts, and is thus worth further studies. Comment: 12 pages, 18 figures, accepted for publication on Astronomy and Astrophysics. Added the subsection "Source Properties" and 3 figures