Integration is introduced as the reversal of differentiation i.e. in solving a differential equation, . The link between integration and area is often passed over and is the subject of the Fundamental Theorem of Calculus. [The following discussion can be adapted for a decreasing function or, piece-wise, a function which successively increases or decreases.]
Consider and area function, , defined by the area under
between
and and a general point,
. If a small increment,
, is applied to
giving a small element,
of area. Now,
dividing though by , gives,
a limit sandwich where, as ,
The curve function, is the derivative of the area function; hence the area function is the anti-derivative of the curve function and,