Put this formula in C1:

=IF(A1=A2,"",B1)

then copy down - it will give you the value for the last occurrence of column A values (assuming they are sorted, as indicated in your example). Then you can just sum column C.

If you want a single formula, then put this array* formula in C1:

=SUM(IF(A$1:A$10<>A$2:A$11,B$1:B$10))

*An array formula needs to be committed using the key combination of Ctrl-Shift-Enter (CSE), rather than the usual <Enter>.

Hope this helps.

Pete