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f(x) = 2π (11 + sin x) √(1 + cos²x)

Step-by-step explanation:

Surface area of a curve rotated about y = a is:

S = ∫ 2π (y − a) ds,

where ds = √(1 + (dy/dx)²) dx.

y = 6 + sin x, and a = -5.  dy/dx = cos x.  Plugging in:

S = ∫₀²ᵖⁱ 2π (6 + sin x − -5) √(1 + cos²x) dx

S = ∫₀²ᵖⁱ 2π (11 + sin x) √(1 + cos²x) dx

Therefore, f(x) = 2π (11 + sin x) √(1 + cos²x).