Question

Find the area under the curve between 10 and 12 for a normal process with mean 6 and standard deviation 4.

Answer 1

N(x) = normal distribution function.
Area under a normal process curve with σ>1, is same as the area under standard normal distribution curve σ=1 after conversion of the variable to standard normal variable. 

The area values are available in a standard table.

$N(x)=\frac{1}{\sqrt{2 \pi} \sigma}\ e^{-\frac{(x-\mu)^2}{2 {\sigma}^2}}\\\\conversion\ z=\frac{x-\mu}{\sigma} \\\\standard\ normal\ distribution\ function=f(z) =\frac{1}{\sqrt{2 \pi}} e^{-\frac{z^2}{2}}\\\\Area\ under\ the\ curve=F(Z)= \int\limits^Z_{-\infty} {f(z)} \, dz$

Normal variable Z for the given variable values 10 and 12:
         Z1 = (10 - 6)/4 = 1            Z2 = (12 - 6)/4 = 1.5

Area under curve between 10 and 12 :  F(Z2) - F(Z1)
     = F(1.5) - F(1.0)
     = 0.93319 - 0.84134 
     = 0.09185
it is 9.185% of the total area under the curve.