Step-by-step explanation:
Graph the region: desmos.com/calculator/rbe6rq61a2
When the region is rotated about y=-2, the resulting shape is a horizontal, hollow cylinder. The volume can be found with either washer method or shell method.
To use washer method, cut a thin vertical slice of the region. Rotated around y=-2, this slice becomes a washer. The width of this washer is dx. The outer radius is 2 − (-2) = 4. The inner radius is y − (-2) = y + 2. The volume of the washer is:
dV = π (4² − (y + 2)²) dx
dV = π (4² − (ln x + 2)²) dx
The total volume is the sum of the washers from x=1 to x=e².
V = ∫ dV
V = ∫₁ᵉ² π (4² − (ln x + 2)²) dx
To instead use shell method, cut a thin horizontal slice of the region. Rotated around y=-2, this slice becomes a cylindrical shell. The thickness of the shell is dy. The radius is y − (-2) = y + 2. The width is x − 1. The volume of the shell is:
dV = 2π (y + 2) (x − 1) dy
dV = 2π (y + 2) (eʸ − 1) dy
The total volume is the sum of the shells from y=0 to y=2.
V = ∫ dV
V = ∫₀² 2π (y + 2) (eʸ − 1) dy
