Final answer:
To find the volume using the cylindrical shells method, we integrate the volume of each infinitesimally thin shell, considering the radius and height function, across the defined limits.
Step-by-step explanation:
Finding the Volume Using the Method of Cylindrical Shells
To find the volume generated by rotating the region bounded by the curves y = 16x⁴, y = 0, and x = 1 about the line x = 2, we use the method of cylindrical shells. The formula for the volume of a cylindrical shell with radius R, height h, and thickness dr is V = 2πRh dr. When rotated about the line x = 2, the radius of the shell becomes R = 2 - x and the height is given by the function h = 16x⁴. The limits of integration span from the y-axis up to x = 1.
The volume V can be found by integrating:
V = ∫(2π(2 - x)(16x⁴)dx)
from 0 to 1.
Carrying out this integration will provide the total volume of the solid of revolution. Remember, the strategy here is not to use the basic volume equations directly, but to sum up the volumes of all infinitesimally thin cylindrical shells.