Can an Arduino board be configured to do summation or integration?
Need to replace an analog integrator (with a time constant of 1 second). The analog integrator is drifting...... can it be replaced with an Arduino
board that does not drift and allows resetting the output to zero with a switch closer.
Yes, probably. There are a bunch of "numerical methods" for doing integration, and full books written on the subject. Basically, you go back to 2nd semester calculus and compute the area of slices of the space under your curve; rectangles using the simpler methods, trapazoids for a slightly more complex method, and so on... Depending on values and timing requirements, you can use either integer or floating point math...
There is a PID control algorythem someone wrote for the arduino.
http://www.arduino.cc/playground/Code/PIDLibrary
There is the sompler expresso library also.
http://www.arduino.cc/playground/Main/BarebonesPIDForEspresso
I think this is what you are looking for.
The analog integrator is drifting.
Note that digital integrators will also drift if the input is noisy or derived from an A/D converter as these are inherently noisy. In theory the noise should cancel but it never seems to do in practice.