Surface Area Integral Question

Question

Find the area of the surface obtained by rotating the curve y = 1 +4x² from x = 0 to x = 5 about the y-axis.

Answer 1

x = √(y - 1)/2; x = 0, y = 1; x = 5, y = 101; dx/dy = 1/[4√(y - 1)].

Surface area A = 2π∫₁¹⁰¹√(y - 1)/2√1 + 1/[16(y - 1)]dy = 2π∫₁¹⁰¹1/16√(8y - 7)dy = π/64(8y - 7)^3/2₁¹⁰¹ =

π/64(807^3/2 - 1) = 1125.2816