Using SUMIF on a Date Range

I'd like to use the SUMIF function to sum data where the date range equals a certain month, is that doable? Here's some sample data:

A B
1/1/2010 5
1/5/2010 2
2/5/2010 7

I want to sum columnB where the month on columnA is 1 or Jan.

Try SUMPRODUCT:

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Cheers,