Find the exact area of the surface obtained by rotating the curve about the x-axis.

Question

y² = x + 1,    0 ≤ x ≤ 15

Answer 1

x = y² - 1, 0 ≤ x ≤ 15; y = ±(1 + x)^1/2 ;

A = 2π∫₀¹⁵[(1 + x)^1/2(1 + 1/[4(1 + x)])^1/2dx = π∫₀¹⁵( 5 + 4x)^1/2 = π/4·2/3(5 + 4x)^3/2₀¹⁵ =

π/6[65^3/2 - 5^3/2) = 268.54