Final answer:
The student is looking for the surface area of a shape formed by rotating the curve x=1/3x^{3/2} about the x-axis, which requires setting up and evaluating an integral from calculus.
Step-by-step explanation:
The student is asking about finding the surface area of the shape obtained by rotating a curve about the x-axis. Here, the curve is described by the equation x=1/3x3/2. Rotating this curve around the x-axis creates a three-dimensional shape, and we can find the surface area of this shape by using integral calculus.
To calculate the surface area of the rotated shape, we need to set up an integral that accounts for the radius at each point of the curve in respect to x. In this case, the outer radius of the yo-yo example and the question about the concave mirror serve as reminders that the radius involved in the calculations will change along the length of the curve.
The exact area of the surface can be found by applying formulas and principles from calculus, which involves setting up and evaluating an integral specifically tailored for surfaces of revolution.