Sumproduct Date Compare for Month and Year

Hello! new here!! I am looking for a formula that will look at the dates in two columns (start and end date), and return the sum of another column if those dates are included in a certain month.

For example below.

The first table, is where my data is stored.

The second table (on another sheet), I want to return the sum of resources needed if the start and end date ranges include that month. So for October in the second table , it would return 2 while November returns 3.

I tried to attach my file, so hopefully it worked.

I have tried this, but am getting a return of 0. It's not calculating how i need it to.

Formula:

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Hope that makes sense and that someone can help me out!

"Where H:I is the start and end date range, and G should be summed..."
That doesn't seem to match the file attached to post #1.
Phuocam's formula seems to sum the "PLANNED APPLICATION ENGINEERS" correctly (i.e. 2 for Oct 18, 2.5 for Nov 18 - March 19 and .5 for April 19).
Let us know if you have any questions.