Yah, so you're missing the main part of your SQL query... (your current query is like a car with no engine)... you need the guts, the part that looks for the dates applicable in the table.
if you're using unix based timestamps while storing your data... you can probably simply add something like so -
SELECT * FROM reports WHERE review = 'reviewed' AND user_name = '$user_name' AND date > '######' AND date < '#####' ORDER BY date DESC
Of course you'll need to create variables to pass to those dates based on what the user selection is, but I think you're only asking about the SQL...
If you need further help (let's say you're not using unix timestamps, but mysql timestamps, or you're stuck on what to feed those pound signs above), drop a line.
good luck