work sb's tail off

MLB レンジャーズのダルビッシュ有投手が、大リーグ３年目で初めての開幕投手を務めることになるようです。

Darvish officially named Texas' Opening Day starter

SURPRISE, Ariz. -- Yu Darvish, leading an international cast of teammates, was told by manager Ron Washington that he will be the Rangers' Opening Day starter on March 31 against the Phillies.

Washington had already said that last month at FanFest, but he told Darvish in a short meeting on Tuesday morning.

"Yu is the kind of guy that doesn't show a lot of emotion, but he was excited," Washington said. "He said he would work his tail off to represent us well and do the job."

http://mlb.mlb.com/news/article/mlb/yu-darvish-officially-named-texas-rangers-opening-day-starter?ymd=20140225&content_id=68311710&vkey=news_mlb

文脈から明らかなように work sb's tail offは「一生懸命に働く、務める」さらには「身を粉にして働く」という意味です。もともとは、tailの箇所にある言葉が用いられていましたが、あまりにも上品さに欠けるため（さて何〜だ？）、その婉曲話法としてtailが用いられるようになったものです。（UG）