This page, like many others on this site, is under major construction. Right now, it pretty much just consists of my lecture notes.

The introductory numerical integration methods are the

Each method can be seen and compared in the Wolfram Demonstrations applet below.

- endpoint methods (left- or right-endpoints),
- midpoint method,
- trapezoid method, and
- Simpson’s method.

Each method can be seen and compared in the Wolfram Demonstrations applet below.

Comparing Basic Numerical Integration Methods from the Wolfram Demonstrations Project by Jim Brandt

The premise of each of the introductory methods lies with the assumption that we have an interval from \(x = a\) to \(x = b\) over which we desire the integral of a function \(f(x)\). In each of these methods we assume the following \[\Delta x = \frac{b - a}{n} \text{ and } x_i = a + i\Delta x\]

**THEOREM | Left-endpoint Method**

\[ \int_a^b{f(x) dx} \approx L_n = \sum_{i = 1}^n{f(x_{i - 1}) \Delta x}\]

**THEOREM | Right-endpoint Method**

\[ \int_a^b{f(x) dx} \approx R_n = \sum_{i = 1}^n{f(x_{i}) \Delta x}\]

**THEOREM | Midpoint Method**

\[ \int_a^b{f(x) dx} \approx M_n = \sum_{i = 1}^n{f(\overline{x}_{i}) \Delta x},\] where \(\overline{x}_i = \frac{x_{i - 1} + x_i}{2}\) is the midpoint of the interval \([x_{i - 1},x_i]\).

**THEOREM | Trapezoidal Method**

\[ \int_a^b{f(x) dx} \approx T_n = \frac{\Delta x}{2}\left(f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n - 1}) + f(x_n)\right)\]

**PROOF | Trapezoidal Method**

All you need to remember is that the area of a trapezoid is \(\frac{1}{2}(h_1 + h_2)w\).

**THEOREM | Simpson's Method**

\[ \int_a^b{f(x) dx} \approx S_n = \frac{\Delta x}{3}\left(f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 4f(x_{n - 1}) + f(x_n)\right),\] where \(n\) is even.

**PROOF | Simpson's Method**

... in class

Each of these methods has an error term associated with it.

**DEFINITION | Numerical error**

If a numerical method is used to approximate the value of an integral, then we say that the

**numerical error**is \[E_{method} = \int_a^b{f(x) dx} - (method)_n,\] where \((method)_n\) denotes the name (and value) of the method used.

We should probably check the errors on these methods and compare each. Let's go to Wolfram Alpha (using this link) and check out, not only the absolute errors between these methods, but what happens when we double the value of \(n\) within each method.

**THEOREM | Error Bounds for the Introductory Numerical Integration Methods**

For trapezoidal and midpoint methods, suppose that \(\forall x \in [a,b], |f''(x)| \leq K\). Then \[|E_T| \leq \frac{K(b - a)^3}{12n^2}\] and \[|E_M| \leq \frac{K(b - a)^3}{24n^2}.\] For Simpson's method, suppose that \(\forall x \in [a,b], |f^{(4)}(x)| \leq K\). Then \[|E_S| \leq \frac{K(b - a)^5}{180n^4}\]

Proofs of these bounds are left for advanced numerical analysis courses.