Try and visualise this (it might help if you draw diagrams):

- You have some curve f(x) and you want to find the area under a small section of the curve at point (a, f(a))

- Then, because this area is small, it can be approximated by the rectangle

- Now imagine tht you have a curve g(x), and you want to find the small change in the value of the function when you have a small change in x,

, at the point (a, g(a)).

-Since the change in x is small, the small change in the function value will approximately equal the gradient at the point a multiplied by the small change in x (approximating the curve with a line).

- That is

-Now we can see that

area under f(x) at the point a only if

. That is, the area under f(x) is only equal to the change in g(x) if g(x) is the anti-derivative of f(x).