Final answer:
To find the area of the surface generated by revolving the curve y = sqrt(2x - x²) about the x-axis, we can use the formula for the surface area of a solid of revolution. This involves finding the limits of integration, substituting the values into the formula, and solving the integral using standard techniques.
Step-by-step explanation:
To find the area of the surface generated by revolving the curve y = sqrt(2x - x²) about the x-axis, we can use the formula for the surface area of a solid of revolution:
A = 2π∫[a,b] y(x)√(1 + (dy/dx)²) dx
In this case, the curve is y = sqrt(2x - x²), so we need to find the limits of integration a and b. We can do this by solving the equation 2x - x² = 0.
Solving for x:
2x - x² = 0
x(2 - x) = 0
x = 0 or x = 2
So the limits of integration are a = 0 and b = 2. Now we can substitute these values into the formula and calculate the area:
A = 2π∫[0,2] sqrt(2x - x²)√(1 + (d(sqrt(2x - x²))/dx)²) dx
This integral can be solved using standard integration techniques, and the result will give us the area of the surface generated by revolving the curve about the x-axis.