You have to fix your input in the following way:
In[214]:= NIntegrate[Exp[-(x^2)], {x, 0, 1}, Method -> "TrapezoidalRule"]
Out[214]= 0.746824
In other words, in
NIntegrate[e^(-(x^2)), {x, 0, 1}, Method -> "TrapezoidalRule"]
you have not properly specified the exponent.
Exponential functions can be entered using the key sequence "Esc e e Esc"...
Further, you can use the options MaxRecursion and Points to calculate the integral with different numbers of sampling points.
In[232]:= Table[ NIntegrate[Exp[-(x^2)], {x, 0, 1}, MaxRecursion -> 0, Method -> {"TrapezoidalRule", "Points" -> n}], {n, {20, 200, 2000}}]
Out[232]= {0.746824, 0.746824, 0.746824}
If you want to see the sampling points use:
Needs["Integration`NIntegrateUtilities`"]
Table[ NIntegrateSamplingPoints@
NIntegrate[Exp[-(x^2)], {x, 0, 1}, MaxRecursion -> 0, Method -> {"TrapezoidalRule", "Points" -> n}], {n, {20, 200, 2000}}]