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You will find out if a four foot cantilever requires using 2x10s or even 2x12s and whether they have to be spaced closer at say 12" centers. If you can extend the cantilever by adding joists of the existing size then you will save a lot of time and effort.
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The distance your joists can safely cantilever or overhang a drop beam is determined by the size of the joists, the wood type and grade of the lumber and the spacing between joists. Many building departments limit joist cantilevers to 24" maximum.
Fill in the parts of the formula that are already known. L stands for the span of the beam, in feet. In the example, the beam spans the 10-foot width of the roof. The total load is 4,000 pounds. F stands for the beam's fiber strength in bending, which depends on the species of wood. For the initial calculation, F can be estimated at 1,000.
A southern pine (2) 2x12 beam can span 8 feet when supporting joists spanning a maximum of 14 feet, and it can cantilever 2 feet beyond the posts on each side. With the allowable cantilever of the joists, this beam would support a deck 12 feet by 17 1/2 feet.
The Building Code (IRC 2009 or IRC 2012) allows a 40 ½” cantilever for a live load of 40 pounds per square foot (psf); your engineer may allow more. Upgrade to 2×10 joists, and the Code says you can cantilever almost 5 feet.³