# The Fundamental Theorem of Calculus

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,