If using XL2007 or greater, read up on SUMIFS.
=SUMIFS(I2:I576,H2:H576,">1/1/2009",H2:H576,"<6/30/2009")
if prior to XL2007, you can achieve this result with SUMPRODUCT
like this:
=SUMPRODUCT(--(H2:H576)>1/1/2009,--(H2:H576)<30/6/2009,I2:I576)
great tutorial on SUMPRODUCT here...