The daily integral of photosynthesis for a population of planktonic cyanobacteria in the Baltic Sea is calculated by numerical analysis based on the photosynthesis–irradiance curve of the organism, the vertical distribution of the population and details of the underwater light field. Variations in the light field are calculated from continuous recordings of surface irradiance and measurements of vertical light attenuation, with corrections for losses by reflection at the water surface that depend on the sun's elevation and roughening by wind. Integrals calculated using models in which simplifying assumptions are made about the surface irradiance overestimate the calculated value by 21% on a cloudless day and by 75%, or more, on an overcast day. Even larger discrepancies (>100%) result when the population, which is concentrated in near-surface layers, is assumed to be evenly distributed with depth. Errors of up to 5% result from neglect of surface reflection, and up to 12% from neglect of changes in phytoplankton distribution over 24 h. Details are given of the method of numerical analysis, using standard computer spreadsheets. Versions of the spreadsheet are described in which effects of changes in the vertical distribution of the phytoplankton and light attenuation can be calculated and the effects of water temperature and photoinhibition can be modelled. The method will improve estimates of primary productivity in natural waters. It can also be used for calculating values of the euphotic depth integrated over 24 h. The spreadsheets are available on http://www.bio.bris.ac.uk/research/walsby/integral.htm.