>>12089529I am fairly certain the trapezoids method is actually worse in terms of error lower bound. It is a bit counterintuitive because it seems like the approximated line is closer to the curve, but it becomes apparent when you consider the following:
If you use the trapezoid method to integrate a smooth curve with, say, a positive value, a positive derivative and a negative second derivative, the trapezoid line will always be below the real curve. Whereas the correct rectangle integral
>>12090155 will be sometimes above and sometimes below, reducing the error. 
