[kqr]

To people who find this stuff useful in practise today (and not merely fascinating or useful 50 years ago): what is your line of work?

I have needed to know the values of a few integrals in my job, but I have always ended up with a close enough answer using computational methods. What am I missing by not solving analytically?

[agarttha]

In quantum mechanics, what you can measure experimentally (observables) are given by integrals. You can do the integrals computationally, but then you only have an empirical understanding of how the observables behave when you change some parameter of your experiment.

In our experiments, we need to know how the frequency of an electromagnetic resonator will change when we couple it to a quantum system. We calculate these frequency shifts with integrals. Being able to calculate these integrals analytically for some limiting cases helps us understand the dependence on the parameters. And usually you can patch the limiting cases together and not even have to compute the integrals numerically.

[ErroneousBosh]

I spend a lot of time working with real-world electronics, where a good mathematical background is important to calculate things like component values for a desired behaviour.

But far better is developing a sense of what's "about right".

I have taught people who studied Electronic Engineering "properly" who calculate that the resistors need to be 20.7kΩ and 21.3kΩ for a given circuit and then will go mad scouring Farnell, Mouser et al for those values.

You or I would say "That needs to be a 22kΩ resistor and an 18kΩ resistor in series with a 4.7kΩ pot, because that is going to need adjusted on test because of the tolerances in everything else", wouldn't we?

[srean]

Often times it so happens that the point of interest is not the numerical value of the integral but its behavior at different points in its domain. If I am able to figure out the expression it becomes easier.

To give an example consider the moment generating transform, Laplace transform. Their symbolic expression can be very informative.

Consider the Mercator projection. It was designed without any idea of the closed form of the required integral. It was mostly done by estimate and gut feel. Now that we know the actual form (an entirely serendipitous discovery) we feel more confident that we understand the transform. This part is considerably psychological but not entirely.

Note that when drawing a map in Mercator projection we have to fall back to numerical estimation. But it helps that parts of the transforms are built from functions tha have names, that means we have seen the same functions elsewhere, it instills a sense of familiarity and understanding.

There are way to many functions to name, so the ones we have given names to are a bit special.

[EGreg]

Frustration. A lot of frustration: https://en.wikipedia.org/wiki/Path_integral_formulation