Final answer:
To find the volume of the solid obtained by rotating the region bounded by the given curves about x = 2, we can use the method of cylindrical shells.
Step-by-step explanation:
To find the volume of the solid obtained by rotating the region bounded by the curves about the line x = 2, we can use the method of cylindrical shells. First, let's sketch the region. The region is bounded by the curves y = 27x³, y = 0, and x = 1. The region is between the y-axis and the curve y = 27x³. When we rotate this region about the line x = 2, it will form a cylindrical shell.
To find the volume of the cylindrical shell, we can integrate the function 2πx(27x³) from x = 0 to x = 1. This integral represents the sum of the volumes of all the infinitesimally thin cylindrical shells that make up the solid. Evaluating this integral gives us the volume V of the solid.