Can you try create a calculated field on the column (containing the date value on which you need to limit the year) and limiting this calculated field to contain date values for year 2010 - 2013.
E.g. if YEAR([ExistingDate])>=2010 and YEAR([ExistingDate]) <= 2013 then [ExistingDate] END
Use this new field in your worksheet as a filter instead.
Just to add to Mandeeps solution above
If you add the new field as a filter and only show relevant values.
also add the existing date as a filter removing the years you do not require
this will remove the null valu that would appear in your calculated field filter.
If you need a hand doing this please share a sample workbook