It is possible to calculate mathematically the area under the normal curve between any two values of z. However, tables/software have been developed to give the areas under the normal curve to the left of particular values of z. The function is the probability of Z<z, or P(Z<z). The area between two values z1 and z2 (where z2>z1) is therefore P(Z<z2)-P(Z<z1).
For example, to find the area between z1=1.5, z2=2.5 is P(Z<2.5)-P(Z<1.5) =0.99379-0.93319 =0.06060
(above values obtained by software, such as R) For example, the value P(Z<2.5) can be calculated using P(Z<2.5)=erf(2.5/sqrt(2))/2+1/2 where erf(x) is a mathematical function that does not have an explicit formula (calculated by summation of series, or tabulated).
