Answer:
(a) V = ∫₂⁵ π ((ln(x))² + 14 ln(x)) dx
(b) V = ∫₂⁵ 2π (x − 1) ln(x) dx
Explanation:
We know the region is the area 0 ≤ y ≤ ln(x) from x=2 to x=5.
(a) Revolve around the line y=-7, and we get a hollow cylinder on its side. Slice vertically into thin washers. The thickness of each washer is dx. The inside radius is r = 0 − (-7) = 7. The outside radius is R = ln(x) − (-7) = ln(x) + 7. The volume of each washer is:
dV = π (R² − r²) t
dV = π ((ln(x) + 7)² − 7²) dx
dV = π ((ln(x))² + 14ln(x) + 49 − 49) dx
dV = π ((ln(x))² + 14 ln(x)) dx
The total volume is the sum of all the washers from x=2 to x=5:
V = ∫ dV
V = ∫₂⁵ π ((ln(x))² + 14 ln(x)) dx
(b) Rotate about x = 1, and we get a hollow cylinder standing upright. Slice into cylindrical shells. The thickness of each shell is dx. The radius of each shell is r = x − 1. The height of each shell is ln(x). The volume of each shell is:
dV = 2π r h t
dV = 2π (x − 1) ln(x) dx
The total volume is the sum of all the shells from x=2 to x=5.
V = ∫ dV
V = ∫₂⁵ 2π (x − 1) ln(x) dx