Final answer:
To find the volume of the solid of revolution, use the method of cylindrical shells by integrating the formula V = 2πrhΔx. The limits of integration are from 0 to 2 because the curves intersect at x = 2. Plug in the values and integrate to find the volume.
Step-by-step explanation:
To find the volume of the solid of revolution obtained by rotating the region bounded by the curves y = x³, the y-axis, and the line y = 8 about the x-axis, we can use the method of cylindrical shells.
The volume of a cylindrical shell is given by the formula V = 2πrhΔx, where r is the distance from the axis of rotation to the shell, h is the height of the shell, and Δx is the width of the shell. In this case, the shell has a height of 8 - x³ and a radius of x. The width of the shell can be represented by Δx.
Integration must be done to find the limits of integration for x. Since the regions are bounded by the y-axis, the line y = 8, and the curve y = x³, we need to find the x-values where these curves intersect. By solving the equation x³ = 8, we find that x = 2. Therefore, the limits of integration are from 0 to 2.
Therefore, the volume of the solid of revolution is given by the integral ∫02 2πx(8 - x³) dx.