Why is absolute convergence required?

Question

Definitions try to precisely formalize intuitive ideas. For example, the concept of mathematical expectation attempts to rigorously formalize the idea of quantifying the average outcome of a random variable, considering all possible values and their respective probabilities. So, given a random variable of the continuous type $X$ with probability density function $f$, in the definition of mathematical expectation of $X$ it is required that $$\int_{-a}^{a} |x|f(x) dx$$ converges if $a \rightarrow +\infty$. What reason is there for this requirement?