Final answer:
To find the volume of the solid obtained by rotating the region bounded by y = x³, x = 8, and the x-axis around x = 10, we use the cylindrical shells method and integrate from 0 to 8 to evaluate the volume.
Step-by-step explanation:
To calculate the volume of the solid obtained by rotating the region bounded by the curve y = x³, the line x = 8, and the x-axis about the line x = 10, we use the method of cylindrical shells. The volume of a typical shell with radius r and height h is given by 2πrhΔr, where Δr is the thickness of the shell.
The radius of a shell is the distance from the y-axis to the shell, which is 10 - x, and the height is the value of the function, which is x³. We integrate from 0 to 8, the limits of integration. Therefore, the volume V can be expressed as:
V = ∫08 2π(10 - x)x³ dx.
This integral can be evaluated to find the exact volume of the solid of revolution.