I used Simpson's rule to calculate the area under a standard distribution where $z\gt1$.
This is the integral: $$A=\frac{1}{\sqrt{2\pi}}\int_1^{\infty}e^{\frac{-x^2}{2}}dx$$
I decided to go as far as $z=4$ and to use $9$ points. So I used my calculator to find values for $z=1.0, 1.4, 1.8, 2.2, 2.6, 3.0, 3.4, 3.8, 4.2$
Simpson's rule gave me a result of $0.1586$. Tables give a result of $0.1587$ so I'm pretty happy with this.
I am only familiar with Simpson's rule and the trapezoidal rule. I have heard of Gaussian quadrature and some other rules.
What is the most convenient/accurate/appropriate rule to use on a Gaussian/standard distribution ?