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).