This paper addresses the survey estimation of a population mean in continuous time. For this purpose we extend the rotation sampling method to functional data. In contrast to conventional rotation designs that select the sample before the survey, our approach randomizes each sample replacement and thus allows for adaptive sampling. Using Markov chain theory, we evaluate the covariance structure and the integrated squared error [ISE] of the related Horvitz-Thompson estimator. Our sampling designs decrease the mean ISE by suitably reallocating the sample across population strata during replacements. They also reduce the variance of the ISE by increasing the frequency or the intensity of replacements. To investigate the benefits of using both current and past measurements in the estimation, we develop a new composite estimator. In an application to electricity usage data, our rotation method outperforms fixed panels and conventional rotation samples. Because of the weak temporal dependence of the data, the composite estimator only slightly improves upon the Horvitz-Thompson estimator.