Area under curve within a particular time window
The sampling can be acheived using sample / hold circuit. Since you don't want to hold anything, then you don't need a hold capacitor. Then you pass the output of S/H circuit to the integrator circuit. So youshould have Sample-&gt;Integrate-&gt;Hold (evaluate result)-&gt;Reset,....
S/H

Integrate/Reset

EDIT:
A good example:
https://digi.lib.ttu.ee/i/file.php?DLID=7459&amp;t=1

http://www.ti.com/lit/ds/sbfs003/sbfs003.pdf
Keep in mind that you need an extra inverting op-amp if you need a positive voltge output.