Existing statistical methods in causal inference often assume the positivity condition, where every individual has some chance of receiving any treatment level regardless of covariates. This assumption could be violated in observational studies with continuous treatments. In this paper, we develop identification and estimation theories for causal effects with continuous treatments (i.e., dose-response curves) without relying on the positivity condition. Our approach identifies and estimates the derivative of the treatment effect for each observed sample, integrating it to the treatment level of interest to mitigate bias from the lack of positivity. The method is grounded in a weaker assumption, satisfied by additive confounding models. We propose a fast and reliable numerical recipe for computing our integral estimator in practice and derive its asymptotic properties. To enable valid inference on the dose-response curve and its derivative, we use the nonparametric bootstrap and establish its consistency. The performances of our proposed estimators are validated through simulation studies and an analysis of the effect of air pollution exposure (PM$_{2.5}$) on cardiovascular mortality rates.