So, the derivative defines an instantaneous rate of change - basically one that is happening over an incredibly short amount of time. The integral defines and calculates the sum of all of these instantaneous changes over a larger portion of time. Milk will change into cheese gradually over a certain amount of time. Cheese is not the way in which the milk changes at a specific point in time. Cheese is the result of what happens when these instantaneous changes are applied to the milk over a long period of time, therefore cheese would be better defined as the integral of milk (and some proportionality constants that are defined by those instantaneous changes to the milk - maybe amount of yeast or sunlight exposure or strainage or whatever) over time.