Here's how to solve for the area of the segment:

1. **Find the points of intersection:**

  - Set the equations equal to each other: 4x = x

  - Solve for x: 3x = 0 => x = 0

  - Substitute x = 0 into either equation to find y: y = 0

  - The point of intersection is (0, 0).

2. **Sketch the curves:**

  - The curve y = 4x is a straight line passing through the origin with a steeper slope than y = x.

  - The line y = x also passes through the origin with a slope of 1.

3. **Set up the integral:**

  - The area of the segment is the area under the curve y = 4x from x = 0 to the point of intersection minus the area under the curve y = x from x = 0 to the point of intersection.

  - This can be represented by the integral: 

    ∫[from 0 to 0] (4x - x) dx 

4. **Evaluate the integral:**

  - ∫[from 0 to 0] (3x) dx = [3x²/2] from 0 to 0 = (3 * 0²/2) - (3 * 0²/2) = 0

5. **The area of the segment is 0 square units.**

**Note:** This result might seem counterintuitive, but it makes sense because the line y = x cuts off a portion of the curve y = 4x that has zero area. This is because both lines pass through the origin, and the segment they create is just a single point.