Question

How to find the area under the standard normal curve between z 1.5 and z 2.5?

Answer 1

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).