You don't need graphs or area at all, of course. All you need is to notice that the frac(x/2) expression is a periodic function that is only integrable piecewis

You don't need graphs or area at all, of course. All you need is to notice that the frac(x/2) expression is a periodic function that is only integrable piecewise, and is easy to integrate with a change of variables for each piece or a handwave of such. It's actually gnarly to write out a formal proof as a new student would do (it requires principle of induction to handle all the pieces), but easy for an expert to breeze through as trivial. The makes it a bit of an unfair problem for a students trying to follow the rules of math. This is very common challenge for students making the transition to higher math, when they are taught rigorous proofs but before they learn that professionals mathematicians are rarely rigorous (except when there is disagreement about the truth of an "obvious" claim).