The formula for the Gaussian distribution is:
Where sigma = std deviation and mu = mean
(stolen from wikipedia)
When you are asking for the area, you are integrating this function over the range specified. This integral does not have a "closed form" solution: there is no way to come up with an expression using "normal" math functions like factorial, multiplication, exponentiation, roots, etc. that equals that integral.
It's just like logarithms or trigonometric functions: you can't produce a closed form equation for them using other algebraic functions (you can use infinite series, but that's not "closed"). So you use a table (if you are feeling retro, or a calculator, which simply uses a table for you behind the scenes embedded in its processor as a starting point) when you need to actually calculate it.
In fact, the parallel with logarithms is quite apt: one can also define a logarithm by an integral, namely ln(x) = integral of (1/x) from 0 to x.