= Solution
Take a collar $\partial_iX\times[0,\varepsilon]$. In the quotient, the two annuli $A_1,A_2$ are folded together by $f$, while their common boundary curves $\beta_1,\beta_2$ become the boundary of a meridional disk. Because $f(\alpha_1)=\alpha_2$, the two halves of $\alpha$ join across this disk and bound it. The quotient of the collar is therefore a solid torus whose meridian is attached along $\alpha$; outside the collar nothing changes. Hence the <annulus-quotient model of Dehn filling> gives
$$
\boxed{X/(A_1\sim_f A_2)\cong X(\alpha).}
$$
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