Averaging switch sensor value
What is V1 exactly, does it have any internal resistance? Does the switch has any internal resistance? 

I think you may have two choices:


Since the switch only changes at 10Hz. You may sample the signal with Arduino ADC at 100Hz, which is easy for Arduino to achieve. Then you can implement averaging algorithm in your Arduino. Say you need the readings every one second, then you can average 100 points every second, using a timer.
Do it in analog way, like yours. You may need a voltage follower, which has low output resistance. The output resistance of your current circuit is high for the input resistance of an ADC. And you may get an inaccurate result. 
This is a voltage follower: http://www.learningaboutelectronics.com/Articles/Voltage-follower