I mean no one integrates to find the area under subsections of the normal distribution because you have to use numerical methods for that anyway. And it only gets more complicated from there with chi-squared and various degrees of freedom etc. Tables are a perfectly fine way to deal with distributions…
Most real life integrals and differential equations are not analytically solvable anyway, so you run into tables like you do with most real life applications of logarithms.
What happens is you learn numerical approximation methods and then recreate the tables with human 'computers', like we used to do before electrical computers came into play. Or create mechanical analog computers like they did in the 1800s or greek times.
That’s question is akin to someone saying what happens when you don’t have a calculator. Everyone has one in their pocket at all times.
How would the typical person go about calculating the normal distribution by hand? Are we going to call everyone blind because they haven’t memorized e to arbitrary precision?
In the land of the blind the guy with one eye walks everyone off a cliff.