Senior secondary students cover numerical integration techniques in their mathematics courses. In particular, students would be familiar with the "midpoint rule," the elementary "trapezoidal rule" and "Simpson's rule." This article derives these techniques by methods which secondary students may not be familiar with and an approach that undergraduate students should be familiar with. Secondary students will also find interesting the "two-point Gauss rule," which is an extension of the trapezoidal rule. There are many applications of integral calculus and developing a deeper understanding of some of the numerical methods will increase understanding of the techniques. The methods chosen in this article have been investigated as secondary students will be familiar with their applications. However, secondary text books and teachers may not use the techniques covered in this article, and this alternate approach may increase the understanding of the importance and applications of the techniques, as well as increase an appreciation of the beauty of mathematics in general. The article also provides a detailed summary of the techniques that will be beneficial for undergraduate students. In this article the midpoint rule, the elementary trapezoidal rule, the two-point Gauss rule and Simpson's elementary rule are developed. (Contains 1 figure.)