Finite Differences and Newton-Cotes integration formulas

Chapter 2 was all about numerical linear algebra where we learnt the techniques of solving linear system of equations using both direct and iterative methods. Chapter 3 is all about finite difference approximation of derivatives which commonly arise in many daily life. We looked into forward, backward and central difference approximation of the derivatives. The forward and backward difference methods are $O(\Delta x)$ while central difference approximation is $O(\Delta x^2)$. For better accuracy of approximation, I would recommend to use the central difference method.
Integration is the process of measuring the area under a function plotted on a graph. The question one would like to know is Why would we want to integrate a function? Among the most common examples are finding the velocity of a body from an acceleration function, and displacement of a body from a velocity function.
Numerical integration is also an important technique. We started off with the simplest method called "Trapezoidal rule" which approximates the integral $I= \int_a^b f(x) dx $ where $a$ and $b$ are lower and upper bounds of the integral through a polynomial of order 1 i.e a linear approximation. The Simpson's $\frac{1}{3}$ rule and $\frac{3}{8}$ rule approximates using a quadratic and cubic polynomial. The main disadvantage of $\frac{1}{3}$ rule is that the number of intervals (n) should always be "even". If "n" is not even, I would either use Trapezoidal rule or Simpson's $\frac{3}{8}$ rule. Trapezoidal and Simpson's rule are all part of Newton-Cotes integration formulas. We shall also talk about "Two-point Gauss Quadrature rule" which is an extension of "Trapezoidal rule" where the arguments of the function are not predetermined as $a$ and $b$ , but as unknowns $x_1$ and $x_2$
$$I= \int_a^b f(x) dx=c_1 f(x_1)+c_2 f(x_2)$$
We shall derive the formula in the lecture on Friday