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The usual way of numerically integrate a function, is to find a simpler function which approximates the given function and then integrating the interpolation function. That is, if we want to find $\int_a^b f(x)\,dx$ , we find an approximating function $p(x)$ such that $f(x)$ and $p(x)$ be close (on some concept of distance) and then we say$$ \int_a^b f(x)\,dx\approx \int_a^b p(x)\,dx$$
The simplest approximation functions are polynomials. If we evaluate $f(x)$ at some points $x_0,x_1,\ldots,x_n$ , we can use Lagrange's interpolating polynomial to find a polynomial $p(x)$ with degree $n$ such that $p(x_j)=f(x_j)$ for $j=0,1,\ldots,n$ .
Newton and Cotes' integration formulas are obtained when the $x_0,x_1,\ldots,x_n$ are sampled evenly over the interval, and then Lagrange interpolating polynomials are used to approximate the function.
The Newton and Cotes formulas for small values of $n$ are given on the following table.
| ${n}$ |
${\int p(x)}$ |
Name |
| $1$ |
$\frac{h}{2}(f(x_0) + f(x_1))$ |
Trapezoidal rule |
| $2$ |
$\frac{h}{3}(f(x_0) + 4f(x_1) + f(x_2))$ |
Simpson's rule |
| $3$ |
$\frac{3h}{8}(f(x_0)+3f(x_1)+3f(x_3)+f(x_3))$ |
Simpson's 3/8 rule |
| $4$ |
$ \frac{2h}{45}(7f(x_0)+32f(x_1)+12f(x_2)+32f(x_3)+7f(x_4))$ |
Milne's rule |
recalling that $x_0,x_1,\ldots,x_n$ are evenly spaced on $[a,b]$ .
Since the Simpson's rule is actually the Newton and Cotes formula for $n=2$ , the proof of Simpson's rule illustrates this method.
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