Pretty much every webmaster runs into this once a year. Someone publishes something on the Web and they did not mean to put it up on the Web. Google crawls it and someone finds this information while searching on Google. The content is now easily accessible to the world. So you take down the page, but it is still in the Google cache and you need it removed asap.
That form specifically states:
If you've worked with the site's webmaster to have the live content changed, and you need us to expedite the removal of our cached copy, please provide us with the URL(s). We'll follow up with you only if we require more info or we have additional info to share.
I never gave it a try myself, but I would love to hear the response time on that form.
Forum discussion at Google Groups.