How do you do b and c?

(a) V = ∫₂⁵ π ((ln(x))² + 14 ln(x)) dx

(b) V = ∫₂⁵ 2π (x − 1) ln(x) dx

Step-by-step 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