Final answer:
To find the volume of the solid obtained by rotating the region bounded by the given curves, use the method of cylindrical shells.
Step-by-step explanation:
To find the volume V of the solid obtained by rotating the region bounded by the curves y = 8x³, y = 8x, x ≥ 0 about the x-axis, we can use the method of cylindrical shells. The volume V can be calculated using the formula V = ∫(2πx · (f(x) - g(x))) dx, where f(x) and g(x) are the functions that bound the region to be rotated.
In this case, f(x) = 8x³ and g(x) = 8x. Substituting these into the formula, we have V = ∫(2πx · (8x³ - 8x)) dx. Solving this integral will give us the volume of the solid.