The method we have for calculating the area or the definite integral, in this case, is still very complicated, as we saw in the previous example, because we will find much worse sums.
To do this, consider the area of the figures when moving the right edge:
If the area is given by A (x), then A (a) = 0, since there is no area at all. Already A (x) gives the area of figure 1, A (b), the area between that is:
that is, A (x) is one of the antiderivatives of f (x). But we know that if F (x) is any antiderivative of f (x), then A (x) = F (x) + C. Doing x = a gives: A (a) = F (a) + C = 0 (A) = 0)
Thus, C =  F (a) and A (x) = F (x)  F (a).
Therefore:

or yet,

Examples:
Note that we have found a way to calculate definite integrals and areas without calculating complicated sums and using only the nonderivative ones.