The integral of a function $f\left(x\right)$ between the points $a$ and $b$ is denoted by

${\int }_a^{b}f\left(x\right) \mathrm{dx}$

and can be roughly described as the area below the graph of $y\=f\left(x\right)$ and above the $x$-axis, minus any area above the graph and below the $x$-axis, and all taken between the points $a$ and $b$.

The integral is important because it is an antiderivative for the original function, that is, if

$g\left(t\right)\={\int }_a^{t}f\left(x\right) \mathrm{dx}$

then

    $g\'\left(x\right)\=f\left(x\right)$.

A Riemann sum is an approximation to the integral, that is, an approximation using rectangles to the area mentioned above. The line segment from $x\=a$ to $x\=b$ is split into $n$ subsegments which form the bases of these rectangles, and the corresponding heights are determined by the value of $f\left(x_i\right)$ at some point $x_i$ between the endpoints of the subsegment. The division of the segment $\left[a\,b\right]$ into subsegments is called a partition. For the sake of convenience we will assume here that the subsegments are of equal width, although this is not strictly necessary.

The Riemann Sum is then given by the general formula:

$\sum _{i\=1}^n f\left(x_i\right) \cdot \left(\frac{b-a}{n}\right)$

There are five main types of Riemann Sums, depending on which point $x_i$ is chosen to determine the height:

Instead of approximating the area under a curve using rectangles or trapezoids, parabolas can be used to approximate each part of a curve.

Let us compute the area under a parabola of the equation $y \= a\cdot x^{2 }\+ b\cdot x \+ c$ passing through the three points $\left(-h\,y_0\right)\,\left(0\,y_1\right)\, \left(h\,y_2\right)$:

As the points $\left(-h\,y_0\right)\,\left(0\,y_1\right)\, \left(h\,y_2\right)$are on the parabola, we have:

Hence:

Therefore, the area under a parabola can be written as:

Hence by adding all the areas under each parabolic arc using three points we can derive:

${\int }_a^{b}f\left(x\right)\mathrm{dx} \= \frac{\mathrm{\Δx} }{3}\left(y_{0 }\+4\cdot y_1 \+ y_2 \right) \+ \frac{\mathrm{\Δx} }{3}\left(y_2\+4\cdot y_3 \+ y_4 \right) \+..\.\+\frac{\mathrm{\Δx} }{3}\left(y_{n-2}\+4\cdot y_{n-1} \+ y_n \right)$

The above equation can be simplified into Simpson's rule:

${\int }_a^{b}f\left(x\right)\mathrm{dx} \approx \frac{\mathrm{\Δx}}{3} \left(y{{}_0\+}_{ }4\cdot y_1 \+2\cdot y_2\+4\cdot y_3\+2\cdot y_4\+4\cdot y_5\+..\+4\cdot y_{n-1}\+y_n\right)$