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In this work, we consider different numerical methods for the approximation of definite integrals. The three basic methods used here are the Midpoint, the Trapezoidal, and Simpson's rules. We trace the behavior of the error when we refine the mesh and show that Richardson's extrapolation improves the rate of convergence of the basic methods when the integrands are sufficiently differentiable many times. However, Richardson's extrapolation does not work when we approximate improper integrals or even proper integrals from functions without smooth derivatives. In order to save computational resources, we construct an adaptive recursive procedure. We also show that there is a lower limit to the error during computations with floating point arithmetic.

This study investigated the annual and seasonal variations in nutrient loads (NO2− + NO3− and orthophosphate) delivered to the Gulf of Mexico from the Mississippi-Atchafalaya River Basin (MARB) and examined the water quality variations. The results indicate that (1) annually, the mean NO2− + NO3− and orthophosphate loads showed a steady increase during 1996–1999, a persistent level during 2000–2007, and a moderate increase during 2008–2016; (2) seasonally, NO2− + NO3− and orthophosphate in MARB in spring and summer were higher than those in autumn and winter. Analysis of variance (ANOVA) identified highly significant differences among seasonal loads; and (3) the median value of NO2− + NO3− in normal weather conditions were higher than that during and right after the hurricanes, while the median value of orthophosphate loads in normal weather conditions was higher than that during the hurricanes, but higher than that right after hurricanes. The two-sample t-test indicates a significant difference (p < 0.046) in orthophosphate loads before and after Hurricane Katrina. Moreover, it is found that there is a significant (p < 0.01) increase in nutrient loads during normal weather conditions. The results indicate that hurricane seasons can significantly influence the nutrient loads from the MARB to the Gulf of Mexico.