Assuming dates are in A2:A12, amounts in B2:B12, and first date from in D3, and first date to in E3, then:
=SUMPRODUCT(($A$2:$A$12>=$D3)*($A$2:$A$12<=$E3)*$B$2:$B$12)/SUMPRODUCT(($A$2:$A$12>=$D3)*($A$2:$A$12<=$E3))
EDIT: Other solutions are more elegant!
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