Let’s have a look at an especially nice function:

f[x_]:=Exp[x]
Plot[f[x],{x,0,1},PlotRange->{0,3}]
integral=Integrate[f[x],{x,0,1}]

Compare the five methods on a single “panel”: 
That is, n=1: just one subinterval.

lrr=1.0
rrr=f[1.0]
trap=(lrr+rrr)/2
mid=f[0.5]
simp=(2*mid+trap)/3

Compare the errors to their bounds of three methods:

This function makes our error analysis especially easy, since f is its own derivative, second derivative, fourth derivative, etc.! The function is also monotone increasing, so the largest value of the derivativess will always be on the right endpoint. 

This will not work in general: You have to study the derivatives over the interval of interest, to find out where they take their largest absolute values (the K in the error formulas).

trapError=Abs[trap-integral]
trapBound=(f[1.0](1-0)^3)/(12*1^2)

midError=Abs[mid-integral]
midBound=(f[1.0](1-0)^3)/(24*1^2)

simpError=Abs[simp-integral]
simpBound=(f[1.0](1-0)^5)/(180*2^2) (* Note the 2^2 in the denominator*)