Maybe this is a stupid question but you're /sci/ so you're probably the only ones who can help me

Consider the following:
You get 2 apples per day, this means your change over time is 2. In other words
dy/dx = 2 and when we integrate this we get y(x) = 2x (with boundary condition y(0) = 0).
So if we want to know how many apples you have at the day 3 you just check the integral of dy/dx which is y and see that y(3) = 6. This makes sense, the area beneath the graph of dy/dx is 2*3 = 6 apples and just logically thinking, you'd get to 6 apples.

However let's take the following
dy/dx = 2x, so every day you get two*the number of days in apples. So day 1 you get 2 apples, day 2 4 apples, etc.
Now if we integrate this, we get y(x) = x^2 (again boundary condition y(0) = 0).

But if we now check how many apples we should get at day 3 it's 0 (day 0) + 2 (day 1) + 4 + 6 = 12 apples.

But y(3) = 9. What the fuck am i doing wrong here??

I realised after some thinking that the correct formula is (x+1)/x*y(x) = x*(x+1) so that you get 3*(3+1) = 3*4 = 12 apples.
Why is the correct formular x*(x+1) and not y(x)??

Am I retarded? Shouldn't the area beneath the graph give me the total amount of change for the number of apples over time. But even if I do that, I get the area of the triangle: 0.5*3(x-axis, days)*6(y-axis, change in apples) = 9.
So the integral gets you the area of the change, but what does this change represent if it's not the total number of apples?

Please help, I got an A for calculus and am doing an engineering Msc but apparently I'm too much of a brainlet to solve this shit..