Final answer:
The volume V of the solid is found by integrating using cylindrical shells, considering the distance to the line x = 2 and the height given by y = 8x³ from x = 0 to x = 1.
Step-by-step explanation:
The question asks to find the volume V of a solid created by rotating a region around a line. The given region is bounded by the curves y = 8x³, y = 0, and x = 1, and the solid is obtained by rotating this region about the line x = 2. To solve this, we use the method of cylindrical shells whose formula is V = ∫ 2πrh dr, where r is the distance to the axis of rotation (which is x = 2 in this case), and h is the height of a shell, given by the function y = 8x³.
To calculate the volume, we integrate from x = 0 to x = 1, and our radius r will be 2 - x because it's the distance from each x to the line x = 2. Therefore, the integral becomes V = ∫ 2π(2 - x)(8x³) dx from x = 0 to x = 1. This integral can be simplified and solved to find the final volume.