Let {{MATH_45}} denote the amount of some substance/population at the time {{MATH_46}}. Assume that the function {{MATH_47}} is continuous, differentiable and its derivative, {{MATH_48}}, continuous on some time interval {{MATH_49}}. Then, the {{CMD_50}}net change{{CMD_51}}, {{MATH_52}}, in the amount {{MATH_53}} over the time interval {{MATH_54}} is given by {{MATH_55}} {{CMD_56}} {{CMD_57}} This comes from the Second Theorem of Calculus. It is clear that {{MATH_58}}, which gives the change in the amount {{MATH_59}} over the time interval {{MATH_60}}. It is the difference between the amount at the end of the time interval and the amount at the beginning of the time interval. The right hand side of the equation, the definite integral, gives the ``accumulation of a rate". You can think of this definite integral as the limit of Riemann sums, where each Riemann sum is the ``sum of the changes of the amount over small intervals of time". {{CMD_61}}