SUMIF(S) with date criteria

I have a 'Detail' worksheet that I am trying to summarize on the 'Recap' worksheet based on Date and Category.

-It's a large worksheet that is continually added to so i would like it to update as I enter new information
-I would like to use the month() formula as one of the criteria as opposed to a ">=" & "<=" date criterias
-With a SUMIF I can get the working Total without a date criteria, but my SUMIFS does not return the correct amount
=SUMIF('Detail'!$E:$E,"A",Detail'!$D:$D)
=SUMIFS('Detail'!D:D,'Detail'!E:E,"A",'Detail'!A:A,"month(1)")

I have attached a sample sheet to show what it is I'm looking for. Thanks so much in advance!

You can't with sumifs.
You'd only be able to do the 2 criteria of >= and <=

You could use sumproduct instead though.

=SUMPRODUCT(--('Detail'!E:E="A"),--(MONTH('Detail'!A:A)=1),'Detail'!D:D)

And even though it's allowed in XL2007+, I highly recommend NOT using entire column references like A:A.
Use specific row #s like A2:A1000