When doing problems, I have noticed the answer key oftentimes uses symmetry (middle point of interval for piecewise function) to deduce the $E(X)$ value, but other times, it'll use the proper formula with integration. Is there an indicator to know when to use which? Thanks. :)
Context: The time it used "symmetry", the function was $\frac{1}{2a}$ in an interval from a to $3a$, so it just said "$E(X) = 2a$ by symmetry".
If the distribution is symmetric, then the expectation, if it exists, is the midpoint. As long as you have shown the distribution is symmetric and the expectation exists, then it is always appropriate.
But a textbook might want to use other techniques even in simple, symmetric cases in order to demonstrate and practice other techniques. Don't expect the textbook to always use the most appropriate technique for every single example problem.
It is also possible that using symmetry and locating the midpoint isn't the easiest way to reach an answer. There is no way to tell generally when this would be the case. Only experience and a trained gut instinct will help here.