Given:
- Radius of the cylindrical conductor = 2 mm = 0.002 m
- Current density varies with the distance from the axis according to J = 103e^(-400r) A/m^2, where r is the distance from the axis in meters.

To find:
The value of the total current (I) in mA.

Explanation:
The total current (I) can be found by integrating the current density over the entire cross-sectional area of the cylindrical conductor.

Step 1: Finding the limits of integration:
Since the current density varies with the distance from the axis, the limits of integration will be from 0 to the radius of the conductor (0.002 m).

Step 2: Setting up the integral:
The current (dI) flowing through an infinitesimally small area (dA) located at a distance (r) from the axis can be given by dI = J * dA, where J is the current density.

Substituting the given expression for current density, we have dI = 103e^(-400r) * dA.

The total current (I) can be obtained by integrating the above expression over the entire cross-sectional area (A) of the conductor.

I = ∫[0 to 0.002] 103e^(-400r) * dA

Step 3: Evaluating the integral:
Since the conductor is cylindrical, the cross-sectional area (dA) can be given by dA = 2πr * dr.

Substituting the values in the integral equation, we have:
I = ∫[0 to 0.002] 103e^(-400r) * 2πr * dr

Integrating the above expression, we get:
I = [-51.5e^(-400r)πr^2] evaluated from 0 to 0.002

Evaluating the integral at the limits, we get:
I = [-51.5e^(-400 * 0.002) * π * (0.002)^2] - [-51.5e^(-400 * 0) * π * (0)^2]

Simplifying the above expression, we get:
I ≈ -51.5e^(-0.8) * π * (0.002)^2

Step 4: Converting the current to mA:
The above value of I is in Amperes. To convert it to milliamperes, we multiply it by 1000.

I ≈ -51.5e^(-0.8) * π * (0.002)^2 * 1000

Using a calculator, we find that I ≈ 7.51 mA.

Conclusion:
Therefore, the value of the total current (I) is approximately 7.51 mA.