The selective classifier (SC) has garnered increasing interest in areas such as medical diagnostics, autonomous driving, and the justice system. The Area Under the Risk-Coverage Curve (AURC) has emerged as the foremost evaluation metric for assessing the performance of SC systems. In this work, we introduce a more straightforward representation of the population AURC, interpretable as a weighted risk function, and propose a Monte Carlo plug-in estimator applicable to finite sample scenarios. We demonstrate that our estimator is consistent and offers a low-bias estimation of the actual weights, with a tightly bounded mean squared error (MSE). We empirically show the effectiveness of this estimator on a comprehensive benchmark across multiple datasets, model architectures, and Confidence Score Functions (CSFs).