Assuming your data setup looks like this:
In cell B6 is the drop down for FY Start and in B7 is the drop down for FY End. The sum of the various rows (X, Y, and Z) for the start - end FY's selected are in cells B9 and down. The formula in cell B9 and copied down is:
=IF(COUNTBLANK($B$6:$B$7)>0,"",SUM(INDEX($B2:$H2,MATCH($B$6,$B$1:$H$1,0)):INDEX($B2:$H2,MATCH($B$7,$B$1:$H$1,0))))