Mapping with unmanned aerial vehicles (UAVs) is a crucial method for acquiring geo-environmental information and conducting environmental monitoring. However, UAVs are constrained by their energy sources, which limits their operational time. Optimizing system performance is a critical challenge that needs to be addressed. This paper investigates a scenario involving cooperative area mapping by a UAV swarm. It introduces a joint optimization model for flight trajectory and computational offloading, aimed at reducing the energy consumption associated with mapping tasks. Specifically, it considers the UAV flight energy model, computational energy model, and offloading strategy. The optimization objective is to minimize the energy usage of the UAV swarm system, utilizing the particle swarm optimization algorithm for the solution. Simulation results demonstrate that the joint optimization scheme is more efficient and outperforms the comparison scheme. Therefore, this method can significantly reduce the energy consumption of the UAV swarm system.