## Rectangular Integration (a.k.a. The Midpoint Rule) – Conceptual Foundations and a Statistical Application in R

#### Introduction

Continuing on the recently born series on numerical integration, this post will introduce rectangular integration.  I will describe the concept behind rectangular integration, show a function in R for how to do it, and use it to check that the $Beta(2, 5)$ distribution actually integrates to 1 over its support set.  This post follows from my previous post on trapezoidal integration.

Image courtesy of Qef from

#### Conceptual Background of Rectangular Integration (a.k.a. The Midpoint Rule)

Rectangular integration is a numerical integration technique that approximates the integral of a function with a rectangle.  It uses rectangles to approximate the area under the curve.  Here are its features:

• The rectangle’s width is determined by the interval of integration.
• One rectangle could span the width of the interval of integration and approximate the entire integral.
• Alternatively, the interval of integration could be sub-divided into $n$ smaller intervals of equal lengths, and $n$ rectangles would used to approximate the integral; each smaller rectangle has the width of the smaller interval.
• The rectangle’s height is the function’s value at the midpoint of its base.
• Within a fixed interval of integration, the approximation becomes more accurate as more rectangles are used; each rectangle becomes narrower, and the height of the rectangle better captures the values of the function within that interval.