15.5 Antiderivates and Areas

In this section we shall assume that we know intuitively what the area of a planar set is. We denote by I the set of all interior points of an interval I; for example [0, ?[ =]0, ?[, [0, 1] =]0, 1[.

The function F is said to be an antiderivative of f on I if F is continuous on I and F ?( x)= f( x) for every point x ? I . The word primitive is used interchangeably for the word antiderivative. If F and G are antiderivatives of f on I then there is a constant c such that

To prove this consider H= F ?G, then H'=0 on I and consequently H is constant on I and therefore on I.

Let us now consider a function f which is continuous and non-negative on [ a, b] and let F be an antiderivative of f on [ a, b]. Let us denote by A( v) the area of

For h positive, A( v+ h) ? A( v) is the area of the set {( x, y); v ? x ? v+ h, 0 ? y ? f( x)} . Clearly

(15.24)

Since f is continuous...