I'm trying to figure out the integral of
$$f(x,y): ([0,1] \times [0,1]) \rightarrow \mathbb{R}$$
$$f(x,y) = \begin{cases}1 & \text{ for } y>x,\\ 0 &\text{otherwise}.\end{cases}$$
Graphically this seems like it should be $\frac{1}{2}$.
However, I cannot see how the upper and lower Riemann integrals give this.