Final Answer:
The domain of a parameterization for a circle of radius 38, centered at the origin, and traced out once in a clockwise direction in the xy-plane is [0, 2π].
Step-by-step explanation:
A parameterization of a circle in the xy-plane can be represented as x = r*cos(θ) and y = r*sin(θ), where r is the radius and θ is the parameter. For a circle centered at the origin, x = r*cos(θ) and y = r*sin(θ). In this case, the radius is 38, so x = 38*cos(θ) and y = 38*sin(θ).
For the circle to be traced out exactly once in a clockwise direction, the parameter θ should cover the interval that corresponds to one complete revolution in the clockwise direction around the circle. In trigonometry, one full revolution in radians is 2π.
Since the circle is traced out once in the clockwise direction, the parameter θ should range from 0 to 2π to cover the complete circle's path without retracing. This range of θ values, from 0 to 2π, ensures that the entire circle is traversed without repetition, representing a complete revolution in the clockwise direction.
Therefore, the domain of the parameterization for the circle, ensuring it is traced out exactly once in a clockwise direction in the xy-plane, is [0, 2π].