Given information:
- Charge density: 1 unit
- Inner radius: 1 m
- Outer radius: 2 m

To find the potential of the coaxial cylinder, we can use the following formula:

V = (1/4πε₀) * ∫(ρ/r) * dV

Where:
- V is the potential
- ε₀ is the permittivity of free space
- ρ is the charge density
- r is the distance from the center of the cylinder
- dV is the differential volume element

Let's calculate the potential step by step:

1. Find the electric field inside the inner cylinder:
Using Gauss's Law, we can find that the electric field inside the inner cylinder is given by:

E = (ρ * r) / (2ε₀)

2. Calculate the potential inside the inner cylinder:
To find the potential, we need to integrate the electric field over the distance from the center to a point inside the inner cylinder. Since the electric field is constant inside the inner cylinder, the potential can be calculated as:

V_inner = -E * Δr
= -(ρ * r) / (2ε₀) * Δr

3. Find the electric field between the inner and outer cylinders:
Inside the coaxial cylinder, the electric field is constant and given by:

E = ρ / (2ε₀)

4. Calculate the potential between the inner and outer cylinders:
To find the potential, we need to integrate the electric field over the distance from the inner to the outer cylinder. Since the electric field is constant between the cylinders, the potential can be calculated as:

V_outer = -E * Δr
= -ρ / (2ε₀) * Δr

5. Calculate the potential at the outer surface of the outer cylinder:
Since the outer surface of the outer cylinder has zero charge density, the potential at this surface is zero.

6. Calculate the potential difference between the inner surface of the inner cylinder and the outer surface of the outer cylinder:
The potential difference is given by:

ΔV = V_outer - V_inner
= -ρ / (2ε₀) * Δr - -(ρ * r) / (2ε₀) * Δr
= ρ / (2ε₀) * Δr + ρ * r / (2ε₀) * Δr
= ρ * (1 + r) / (2ε₀) * Δr

7. Substitute the given values:
To find the potential difference, we need to substitute the given values into the equation:

ΔV = 1 * (1 + 2) / (2 * 8.85 * 10^-12) * Δr
= 3 / (17.7 * 10^-12) * Δr

8. Calculate the potential:
The potential can be calculated as the negative of the potential difference:

V = -ΔV
= -3 / (17.7 * 10^-12) * Δr

9. Convert the potential into the desired units:
To convert the potential into 10^9 units, we divide the potential by 10^9:

V = -3 / (17.7 * 10^-12 * 10^9) * Δr
= -3 / 17.7 * Δr