Maybe something like this...

Data Range
A
B
C
D
E
F
1
Date
Value
------
Start
End
Total
2
4/27/2006
37
9/25/2000
11/19/2006
234
3
4/4/2011
81
4
7/16/2006
79
5
10/8/2014
5
6
11/11/2004
28
7
8/29/2002
29
8
11/13/2011
58
9
9/10/2002
61
10
11/19/2012
7


This formula entered in F2:

=SUMIFS(B2:B10,A2:A10,">="&D2,A2:A10,"<="&E2)