Consider the integral $$ I = \int_0^a \int_0^b f(x,y)\,dy\,dx. $$
I can, in order to approximate the integral, use some kind of "trapezoidal rule" $$ I \approx I_{\mathrm{tr}} = \frac{ab}4\bigl(f(0,0) + f(a,0) + f(a,b) + f(0,b)\bigr) $$ or use some kind of "midpoint rule" $$ I \approx I_{\mathrm{mp}} = ab\cdot f\left(\frac{0+a}2,\frac{0+b}2\right). $$
Do these rules have specific names, maybe even their $n$-dimensional generalizations?
I think they should be accurate of second order: $$ I - I_{\mathrm{tr}}, I - I_{\mathrm{mp}}\in\mathcal O\bigl((ab)^2\bigr) $$ is there a source that proves it (so I don't have to)?