Flood attributes such as the water level may depend on multiple forcing variables that arise from common meteorological conditions. To correctly estimate flood risk in these situations, it is necessary to account for the joint probability distribution of all the relevant forcing variables. An example of a joint probability approach is the design variable method, which focuses on the extremes of the forcing variables, and approximates the hydraulic response to forcing variables with a water level table. In practice, however, application of the design variable method is limited, even for the bivariate case, partly because of the high computational cost of the hydrologic/hydraulic simulations. We develop methods to minimise the computational cost and assess the appropriate extent and resolution of the water level table in a bivariate context. Flood risk is then evaluated as a bivariate integral, which we implement as an equivalent line integral. The line integral is two orders of magnitude quicker and therefore beneficial to settings that require multiple evaluations of the flood risk (e.g., optimisation studies or uncertainty analyses). The proposed method is illustrated using a coastal case study in which floods are caused by extreme rainfall and storm tide. An open-source R package has been developed to facilitate the uptake of joint probability methods among researchers and practitioners.