Adding up data that corresponds to a particular range of inputted dates using LOOKUP

I have a sheet with a column of dates (ColumnA), and next to it is a second column with data that corresponds to those dates (ColumnB). I have a field for a user to type in a start date (FieldA) and a field for a user to type in an end date (FieldB) and i want it to add up all the data in ColumnB that corresponds to the range of dates from FieldA to FieldB in Cell1. I know how to use LOOKUP to find the data that corresponds to FieldA and the data that corresponds to FieldB, but how do I make it sum up all the data for all the dates in between as well? This was my best guess for code for Cell1:

=SUM(LOOKUP(FieldA, ColumnA, ColumnB):LOOKUP(FieldB, ColumnA, ColumnB))

But that doesn't work. I want to add up all the data that corresponds to an entered date range, and I'm missing something. Thanks!