Final answer:
The volume of the solid created by rotating the region bounded by y=8√x, y=0, and x=1 about the line x=-2 can be calculated using cylindrical shells. Each shell's volume element is given by 2π(x + 2)(8√x)dx. Integrating from 0 to 1 gives the total volume V.
Step-by-step explanation:
To use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the curves y = 8 √x, y = 0, x = 1; about the axis x = -2, we consider a typical shell at a distance x from the y-axis with a height given by y = 8 √x. The volume dV of the cylindrical shell with thickness dx is represented by dV = 2π(x + 2)(8 √x)dx, where (x + 2) is the radius of the shell since it's rotated about x = -2. The total volume V is found by integrating this expression from x = 0 to x = 1.
The integral we need to solve is:
V = ∫01 2π(x + 2)(8 √x)dx.
Executing this calculation yields the volume of the solid of revolution.