Question

State the Midpoint Rule.

Short answer

The Midpoint Rule states that: \(\mathop {lim}\limits_{n \to \infty } \;\sum\limits_{i = 1}^n {f({m_i})} \Delta x = \int\limits_a^b {f(x)\;} dx\)

Step-by-step solution

Step 1: Definition of Midpoint Rule 

The midpoint rule, also known as the rectangle method or mid-ordinate rule, is used to approximate the area under a simple curve. There are other methods to approximate the area, such as the left rectangle or right rectangle sum, but the midpoint rule gives a better estimate compared to the two methods.

Step 2: Assumptions and notations with midpoint definition

Let f(x) be a continuous function on (a,b), n a positive integer and \(\Delta x = \frac{{b - a}}{n}\).

Divide the interval in n subintervals, each of length, \(\Delta x\)and let \({m_i}\)the midpoint of the \({i^{th}}\)subinterval. Set

\({M_n} = \sum\limits_{i = 1}^n {f({m_i})\Delta x} \)

Then we have:

\(\mathop {lim}\limits_{n \to \infty } \;\sum\limits_{i = 1}^n {f({m_i})} \Delta x = \int\limits_a^b {f(x)\;} dx\)