Answer
(a) i) $ \int ^{2}_{1} 2\pi x^{-2} \sqrt{1+4x^{-6}} dx$
ii) $\int^{2}_{1} 2\pi x \sqrt{1+4x^{-6}} dx$
(b) i) 4.4566 ii) 11.7299
Work Step by Step
(a) i) $y = x^{-2}$ then $dy/dx = -2 x^{-3}$ and $ds = \sqrt{1+(dy/dx)^{2}}dx = \sqrt{1+4x^{-6}}dx$
So
$S = \int 2\pi y ds = \int ^{2}_{1} 2\pi x^{-2} \sqrt{1+4x^{-6}} dx$
ii) $S = \int 2\pi x ds = \int^{2}_{1} 2\pi x \sqrt{1+4x^{-6}} dx$
(b) i) 4.4566 ii) 11.7299
