Verified answer

The surface area equals

∫ 2πx ds, since the region is being rotated about the y-axis

= ∫ 2πx √(1 + (dy/dx)^2) dx

= ∫(x = 0 to 2) 2πx √(1 + (-2x)^2) dx, since y = 3 - x^2

= ∫(x = 0 to 2) 2πx (1 + 4x^2)^(1/2) dx

= ∫(u = 1 to 17) 2π u^(1/2) du/8, letting u = 1 + 4x^2, du = 8x dx

= (π/4) * (2/3)u^(3/2) {for u = 1 to 17}

= (π/6) (17^(3/2) - 1).

I hope this helps!