Given a natural number n>0, consider the following dissection of the interval [0,2]: . Consider next these two step functions:, , whose graphs are represented here below, in the case n = 5.
Let's now calculate the integrals of these two step functions . = , where we have used the formula for the sum of an arithmetic progression. Using the same strategy we obtain
If n gets greater and greater these two step functions become closer and closer to f(x)=x, and their integrals tend to the common value 2. It's almost obvious that 2 is the supremum of the integrals of all lower step approximations of f, and, at the same time, the infimum of the integrals of all upper step approximations of f. So . Observe that the area between the function and the x-axis can also be calculated by elementary techniques, with the same value.