Sum certain items for a date range

I have two sheets:

1. Sheet 1: this has the following columns: Date, Category, Cost
2. Sheet 2: Calculates the total for each category with the sumif formula. It has the following columns: Category, Total.

Now I want to add a breakdown of my totals of each category for each month. This would cause Sheet 2 to receive additional columns for each month. Turning it into something like: Category, Total, January, February, March, ...., December.

The problem is that I don't know how to limit the sumif function to a certain date range. Any help would be greatly appreciated.

Thanks Alan! It works perfect now.