ahh ok. i think i can explain it a little bit for you. so we'll try with the simplest example where there is constant acceleration.

so i know you know this but im just going to state it here again. if acceleration is constant, then the velocity will be proportional to t from integrating over a constant. then, with a velocity that is proportional to t, we integrate and get a position function that is proportional to t²

A| | _|----------------------- _| _|_______________t

V| / | / _| / _| /
_|/______________t

x|
| some quadratic shape here _|
_|
_|______________t

for the graphical intuition, we know that integrals over a range from a to b is just the total area under a plot between points a and b.

you can go from top to bottom first by saying" okay. i know that velocity is just adding up all of the areas under the acceleration graph. if the acceleration is constant, then im always going to be adding by that same number. therefore, my velocity is always increasing by a constant amount and it must be a sloped line!"

then we thing "hey the position function is going to be the sum of the area under the velocity plot. but for the velocity plot, as time goes on, the area isnt increasing by a constant amount like in the acceleration plot...im adding more and more area for each increment of time than i was for the previous increment of time. that means my position isnt increasing linearly. its quadratic!"

does that kind of make sense? if you can look at the acceleration plot and determine what kind of function it is, then that's also a pretty quick shortcut too.