Something has me confused and it's probably pretty simple.

If I find the area under some function between two points a and b, the Fundamental Theorem says it's equal to the difference of the functional values of the 'parent' function valued at the same two points. Say f(x) = x then I'm calling its parent F(x) = (x^2)/2, as an example.

If the x and y coordinates refer to lengths, like meters, then the area under the curve would be expressed in square meters. But F(b) - F(a) is also equal to this area. But it's coming from the difference of the functional values found on its y axis.

So how did this y axis get to become units of square meters?

And then I could do it all over starting with the area under f(x) = x^2 and get square meters as the dimension of my answer, and then find the same answer from F(b) - F(a) from F(x) = (x^3)/3?

I must have a flaw in my understanding here!