Banding Variable in R

For the following code:

x <- data.frame(year = c(1730, 1860, 1941, 2011))

century_bands <- data.frame(min_year = c(1700, 1800, 1900, 2000),
                            max_year = c(1799, 1899, 1999, 2099),
                            century_name = c("18th", "19th", "20th", "21st"))

I'd like, for each value in x, to work out the name of the century it falls into, using the information in century_bands. I can't imagine this is difficult to achieve but I can't figure it out. Can anyone help please? Is there a way using the dplyr package (which I use quite a lot) or perhaps some other technique?

This is just a very simple example of a real-life situation where the bands aren't in nice neat 100 year steps - so any shortcuts based on dividing the year by 100 etc. won't work unfortunately.

Thank you.