Question

Question #ffcc3

Answer 1

Lower integral = 2
Upper integral = 4
(Assuming that we are talking about Riemann integrals here)

Explanation:

To work out the lower(upper) Riemann integral,

• the interval of integration is divided into many pieces,
• the lowest(highest) value of the integrand in each piece is multiplied by the size of the interval
• these are added together
• finally a limit is taken where the largest of the pieces shrink to zero.

We know that both rational and irrational numbers are dense on `RR`, thus no matter how small an interval is it will always have a rational value of `x` and an irrational value of`x` (actually the number of points of both kinds are infinite - but for what follows "at least one" is all we need. Thus, the smallest value of `f(x)` in every piece is 1, while the largest value, again in every piece is 2.

So, the lower Riemann integral is the sum of `1 times` the size of the pieces - which is just the size of the interval `[-1,1]`, i.e. 2
Similarly, the upper Riemann integral is twice the width of the interval , i.e. 4

Since the lower and upper Riemann integrals differ - this function is not Riemann integrable.

By the way, it is Lebesgue integrable - with a value of +4.