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Most problems in Applied Mathematics involving difficulties such as nonlinear governing equations and boundary conditions, variable coefficients and complex boundary shapes preclude exact solutions. Consequently exact solutions are approximated with ones using numerical techniques, analytical techniques or a combination of both. This book looks at the low frequency vibrations of a thin shell of revolution with a curvature which changes sign. Integrals of the equilibrium equations and stress-strain relations are represented in the form of asymptotic series and their solutions as a combination of the Airy function and its derivative.