= Solution
For a nonnegative <random variable>, $T=\int_0^\infty\mathbf1\{T>t\}\,dt$. <Tonelli theorem> permits interchanging expectation and this nonnegative integral, giving the <tail integral formula for moments>:
$$
\boxed{\mathbb ET=\int_0^\infty\mathbb P(T>t)\,dt=\int_0^\infty S(t)\,dt.}
$$
The equality also holds when the mean is infinite. Thus mean completion time is the total area under the <survival function>, while a <restricted mean survival time> integrates only to a specified follow-up horizon.
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