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In this article, we study complete surfaces with finite topological type and has at most quadratic area growth. In particular, we show that if the curvature of such a surface does not change sign, then it must be of finite total curvature. Mathematics Subject Classification (1991). 53C20 Keywords. Riemannian manifold, Gaussian curvature, total curvature, finite topological type. In 1935, Cohn-Vossen [CV] studied the validity of the Gauss-Bonnet theorem for complete non-compact surfaces. Indoi:10.1007/pl00000367 fatcat:ulv7ksiktzfwngeyz7p5o6fbrq