The problem states that the current in a conductor varies with time as I = 2t - 3t^2, where I is in amperes and t is in seconds. We need to find the electric charge flowing through a section of the conductor during the time interval from t = 2 seconds to t = 3 seconds.

To find the electric charge, we can integrate the current function with respect to time over the given time interval. The integral of a function represents the accumulated sum of the function over a given range.

Let's solve the problem step by step:

Step 1: Determine the integral of the current function.

∫[2t - 3t^2] dt

To integrate, we can use the power rule of integration, which states that the integral of t^n with respect to t is equal to (1/(n+1)) * t^(n+1). Applying this rule to each term of the current function, we get:

∫[2t] dt - ∫[3t^2] dt

= (2/2)t^2 - (3/3)t^3 + C

Simplifying further:

= t^2 - t^3 + C

Where C is the constant of integration.

Step 2: Evaluate the integral over the given time interval.

To find the electric charge flowing through the conductor during the time interval from t = 2 seconds to t = 3 seconds, we substitute the upper and lower limits into the integral:

Q = [t^2 - t^3] from 2 to 3

= (3^2 - 3^3) - (2^2 - 2^3)

Simplifying:

= (9 - 27) - (4 - 8)

= -18 - (-4)

= -18 + 4

Step 3: Calculate the electric charge.

Q = -14

The electric charge flowing through the section of the conductor during the time interval from t = 2 seconds to t = 3 seconds is -14 Coulombs.

Therefore, the electric charge is negative, indicating that the direction of charge flow is opposite to the conventional direction of current flow.