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Derivative of a signal or function is its instantaneous rate of change. For a differentiator Capacitor connects the load resistor to i/p. Here small value of capacitor ensures that only instantaneous changes of i/p are coupled to the o/p. An eg. is a differentiator is used for generating impulse waveform from a square wave.
Integration is a continuous summing up action over time domain. For that the values of s/l at various time instants must be stored and added with amplitude of s/l at coming instants of time. The values of capacitor must be large to ensure that s/l's values will remain stored for a specified time(depending on the frequency of s/l under consideration) and get added up. This is because larger the value of a capacitor smaller will the decay of charge from capacitor.
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