First off, they're going to want to know why.
Second, they're going to suggest you use any vacation time you have.
Google has a number of sites with correspondence examples to reference.
Personally, I'd just go with the truth.
I really need 10 days off (from when to when) for (reason).
I understand that this may put an extra load on everyone else, and am more than willing to switch shifts or help out in any way I can so that this will work for everyone.
Give them 48 hours, then speak to them personally about it.