Final answer:
To find the exact area of the surface obtained by rotating the curve about the x-axis, use the formula for finding the surface area of a solid of revolution.
Step-by-step explanation:
To find the exact area of the surface obtained by rotating the curve about the x-axis, we need to use the formula for finding the surface area of a solid of revolution. The formula is given by:
Surface Area = 2π ∫a b y √(1 + (dy/dx)²) dx
In this case, the curve is defined by the equation x = (1/3)*(y² + 2)³/² and the limits of integration are 1 and 2. Plug in the values and evaluate the integral to find the exact area.