help with formula to add values up if cell falls within a date range

Hello,

I've exhausted my Excel knowledge. What I'm trying to do is take the values of particular cells and add them together if another cell (in the same row) falls between certain dates. For example:

Cell A1 has the start date (1/1/15) and B1 has the end date (12/31/15)
Cell D1 is where the formula would be that adds up the total
Cell A4 has date 6/10/15 and B4 has the value of 5
Cell A5 has date 1/15/16 and B6 has the value of 7
Cell A6 has date 7/13/15 and B5 has the value of 3
Cell A7 has date 1/20/16 and B7 has the value of 2

What I'm trying to get cell D1 to do is add the cells B4 through B7 together if the cells A4 through A7 fall between the dates of cells A1 and B1.

So, if the dates are 1/1/15 - 12/31/15, the value of D1 would automatically equal 8. And if I changed the year from 15 to 16, D1 would equal 9.

The spreadsheet that I am working with has hundreds of values, rows, etc., but the example above would greatly help me if someone has any thoughts.

Thanks!

I'm using the 2010 version from Office Standard. I also like to export these to Google Docs spreadsheet feature, if that makes any difference.

Thanks. My total cell (D1) comes up with 0. But oddly enough as an experiment, if I changed the dates to numerical values and went about it that way, it works for the range. Do you know of another way to invoke it to read the range of dates? I've tried all sorts of date formatting too (i.e. 1/10/15, 01/10/15, 01/10/2015, etc

I added them as dates (used the Format cell approach to select the date). Is there anything that I'm missing that is making the cell appear as a non-date?

so, I tried it with Excel 2013 and it works per your formula. I suppose there is something wiry with 2010? Anyone, I greatly appreciate your help and the follow through assistance!