Newton-Cotes Formulas are numerical methods used to approximate the definite integral of a function. They work by using polynomial interpolation to estimate the area under a curve. The formulas can be classified into two main types: closed and open. Closed formulas use the function values at the endpoints of the interval, while open formulas do not. These formulas are derived from Isaac Newton and Roger Cotes, who developed them in the early 18th century. They are particularly useful for functions that are difficult to integrate analytically, providing a way to obtain approximate solutions with a manageable level of complexity.