I have some difficulties to imagine how the two cylinders are placed in the cube. I've sketched a possible situation. (see attachment)

If this is correct you have to calculate the painted area which is the base of the volume surrounding the two cylinders.

I consider only the half of the base square of the cube. Then the base circle of one cylinder is the inscribed circle to the isosceles right triangle forming the half of the base square.

Let s denote the side of the square and d the diagonal of the square.

Then

Then the radius of this circle can be calculated by:

Since the radius is:

With s = 2 the radius

The area of the horizontal slice becomes:

With s = 2 the area is

And therefore the volume surrounding the two cylinders is: