Final answer:
The volume of the solid generated by rotating the region bounded by the curves y = x³, y = 8, and x = 0 around the x-axis is π(64/5) cubic units, calculated using the cylindrical shells method.
Step-by-step explanation:
The question asks us to find the volume of a solid generated by rotating the region bounded by the curves y = x³, y = 8, and x = 0 about the x-axis using the method of cylindrical shells. To use this method, consider a typical element at position x with width dx, which forms a cylindrical shell when rotated about the x-axis. The height of this shell is y = x³, and the radius is x. The volume of a thin cylindrical shell is the circumference of the shell (2πx) times the height (x³) times the thickness (dx). Therefore, the formula for the volume of a thin shell is dV = 2πx(x³)dx.
To find the total volume V, we integrate the volume of the shells from x = 0 to the point where y = x³ intersects y = 8, which occurs at x = 2. Hence, the volume of the solid is given by the integral V = ∫2πx(x³)dx from 0 to 2. Performing the integration, V equals 2π times the integral of x⁴ from 0 to 2, which amounts to V = 2π[ frac{1}{5}x⁵]⁻₂₀ = frac{2}{5}π(32) = frac{64}{5}π cubic units