Chapter 12.1, Problem 111E

Surfaces of revolution Let C be the curve x = f(t), y = g(t), for a ≤ t ≤ b, where f′ and g′ are continuous on [a, b] and C does not intersect itself, except possibly at its endpoints. If g is nonnegative on [a, b], then the area of the surface obtained by revolving C about the x-axis is

$S={\displaystyle {\int }_a^{b}2\pi g\left(t\right)\sqrt{f\prime {\left(t\right)}^2+g\prime {\left(t\right)}^2}dt}.$

Likewise, if f is nonnegative on [a, b], then the area of the surface obtained by revolving C about the y-axis is

$S={\displaystyle {\int }_a^{b}2\pi f\left(t\right)}\sqrt{f\prime {\left(t\right)}^2+g\prime {\left(t\right)}^2}dt.$

(These results can be derived in a manner similar to the derivations given in Section 6.6 for surfaces of revolution generated by the curve y = f(x).)

111. A surface is obtained by revolving the curve x = e^3t + 1, y = e^2t, for 0 ≤ t ≤ 1, about the y-axis. Find an integral that gives the area of the surface and approximate the value of the integral.