ARC LENGTH AND SURFACE AREA

Question

Find the area of the surface obtained by rotating the curve y=1+6x^2 from 𝑥=0 to x=4 about the y-axis. 

Answer 1

Calculate using conic shells

f'(x) = 12x

L(x) = √(f'(x)²+1) = √(144x²+1)

S = 2π∫₀⁴xL(x)dx = 2π∫₀⁴x√(144x²+1)dx

let u = 144x²+1, du = 288xdx

when x = 0, u = 1

when x = 4, u=2305

S=(2π/288)∫₁²³⁰⁵u^1/2du = (π/144)(2/3)u^3/2|₁²³⁰⁵ = (π/216)(2305^3/2-1)

Answer 2

Answer 3

x = 0, x = 4, y' = 12x

Surface area S = ∫₀⁴ 2πx(1 + 144x²)^1/2dx = 2π/288·2/3(1 + 144x²)^3/2₀⁴ = π/216(2305^3/2 - 1) = 1609.53