I'm looking for some challenging problems of the type: "Prove this is integrable from scratch" or "Calculate the integral from scratch". For example, I'm looking for for problems to give students roughly of the difficulty of:
Prove $f$ defined by $$f(x) := \begin{cases} 0 & x \, \, \text{irrational} \\ \frac{1}{q} & x = \frac{p}{q} \, \, \text{in lowest terms} \\ \end{cases} $$
is integrable on $[0,1]$ by definition.
Problems can be in more than one dimension, such as proving that the product of two monotone functions of different variables is integrable on a square.