In terms of Riemann integrals:

$$I(a) = \int_{-a}^0 x f(x) dx \\ J(b) = \int_{0}^b x f(x) dx $$

$$ \begin{array}{rccr} E[X] &=& \lim_{(a,b) \to (\infty,\infty)}& I(a)+J(b) \\ E[|X|] &=& \lim_{(a,b) \to (\infty,\infty)} &-I(a)+J(b) \end{array}$$

These double limits are finite if and only if the single limits $ \lim_{a \to \infty} I(a)$ and $ \lim_{b \to \infty} J(b)$ are finite. (And as a consequence $E[X]$ is finite iff $E[|X|]$ is finite)

Note that for the limit $E[X]$ to be finite, it implies that the single terms $I(a)$ and $J(b)$ converge when $a$ and $b$ go to infinity. This is because their sum is bounded for any sufficiently large values.

That is: If $E[X]$ is finite, then for every $\epsilon>0$ there is value $N$ such that for every sufficiently large values, $a>N$ and $b>N$ we have $E[X]-\epsilon \leq I(a) + J(b) \leq E[X]+ \epsilon$.

But such bounds are not possible if $I(a)$ and $J(b)$ diverge (because then we can chooses some $a > N$ and $ b > N$ that make $I(a) + J(b)$ outside the bounds). Therefore $I(a)$ and $J(b)$ must converge if $E[X]$ is finite.

In terms of Riemann integrals:

$$I(a) = \int_{-a}^0 x f(x) dx \\ J(b) = \int_{0}^b x f(x) dx $$

$$ \begin{array}{rccr} E[X] &=& \lim_{(a,b) \to (\infty,\infty)}& I(a)+J(b) \\ E[|X|] &=& \lim_{(a,b) \to (\infty,\infty)} &-I(a)+J(b) \end{array}$$

These double limits are finite if and only if the single limits $ \lim_{a \to \infty} I(a)$ and $ \lim_{b \to \infty} J(b)$ are finite. (And as a consequence $E[X]$ is finite iff $E[|X|]$ is finite)

Note that for the limit $E[X]$ to be finite, it implies that the single terms $I(a)$ and $J(b)$ converge when $a$ and $b$ go to infinity. This is because their sum is bounded for any sufficiently large values.

That is: If $E[X]$ is finite, then for every $\epsilon>0$ there is value $n$ such that for every sufficiently large values, $a>n$ and $b>n$ we have $E[X]-\epsilon \leq I(a) + J(b) \leq E[X]+ \epsilon$.

But such bounds are not possible if $I(a)$ and $J(b)$ diverge (because then we can chooses some $a > n$ and $ b > n$ that make $I(a) + J(b)$ outside the bounds). Therefore $I(a)$ and $J(b)$ must converge if $E[X]$ is finite.