Explanation:
S = ∫ 2π y ds
ds = √(1 + (dx/dy)²) dy
ds = √(1 + (8y)²) dy
ds = √(1 + 64y²) dy
S = ∫₁² 2π y √(1 + 64y²) dy
S = π/64 ∫₁² 128y √(1 + 64y²) dy
S = π/64 [⅔ (1 + 64y²)^(³/₂)] |₁²
S = π/96 (1 + 64y²)^(³/₂) |₁²
S = π/96 (1 + 256)^(³/₂) − π/96 (1 + 64)^(³/₂)
S = π/96 (257√257) − π/96 (65√65)
S = π/96 (257√257 − 65√65)