Final answer:
To find the area of the surface generated by revolving the curve y=x³ with 0 ≤ x ≤ 0.5, we can use the formula for finding the surface area of a solid of revolution.
Step-by-step explanation:
Calculating the Area of the Surface Generated by Revolving the Curve y=x³
To find the area of the surface generated by revolving the curve y=x³ with 0 ≤ x ≤ 0.5, we can use the method of integration. Since the curve is being revolved around the x-axis, we will use the formula for finding the surface area of a solid of revolution:
Surface Area = 2π ∫[a, b] y(x) √(1 + (dy/dx)²) dx
Substituting the equation y=x³ and the given limits 0 and 0.5, the integral becomes:
Surface Area = 2π ∫[0, 0.5] x³ √(1 + 9x⁴) dx
Integrating this equation and evaluating it between the limits will give us the surface area of the solid of revolution.