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.

midpoint rule

Image courtesy of Qef from Wikimedia Commons.

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.

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