Problem

Consider the function

Is it possible to find the area between this function and the x axis? If so, what is this area?

Solution

The area under a curve and above an interval is defined as the common limit of the Upper and Lower Riemann Sums (the limit as the maximum width of the subintervals goes to 0 and the number of subintervals goes to infinity). If the Upper and Lower Riemann Sums of a function over an interval approach different values, we can't find a value for the area under the graph of that function over the interval.

This said, let's examine the Upper and Lower Riemann Sums of f. First, we'll subdivide [0, 1] into n subintervals, labelling the endpoints

This is our partition P.

Now let's calculate U(P, f). For the Upper Riemann Sum, remember that the height of each rectangle is the maximum value M_i of the function on that ith subinterval. So

A property of the rational numbers is that any interval of real numbers, no matter how small, contains a rational number. so M_i = 1 for all i, since 1 is the function's maximum value. Therefore

since the interval has length 1. We can take Dx_i as small we we want, and large as we want, we'll still get 1. So, if the region has an area, it should be equal to 1.

Now let's find L(P, f). For the Lower Riemann Sum, the height of each rectangle is the minimum value m_i of the function on that ith inteval. So

A property of the irrational numbers is that any interval of real numbers, no matter how small, contains an irrational number. So m_i = 0 for all i, since 0 is the function's minimum value and every interval contains an irrational number. Thus L(P, f) = 0 for any partition P.

So the Upper and Lower Riemann Sums for f(x) do not approach a common value, and we can't define the area between this function and the x axis.