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 using rectangles to approximate the area under the curve, trapezoids give a better approximation to the area.

The area of a trapezoid with base $b$ and heights $p$ and $q$ is given by:

$\mathrm{Area} \= \frac{b}{2}\left(p\+q\right)$

To approximate the area under the curve, add up the area of all the trapezoids.

$\mathrm{Area} \=\frac{1}{2}\left(y_0 \+ y_1\right)\mathrm{\Δx} \+ \frac{1}{2}\left(y_1 \+ y_2\right)\mathrm{\Δx} \+ \frac{1}{2}\left(y_2 \+ y_3\right)\mathrm{\Δx} ..\.$

which can be simplified to become the trapezoidal rule:

$\mathrm{Area} \approx \mathrm{\Δx}\left(\frac{y_0}{2} \+ y_1\+ y_2 \+ y_3\+..\+\frac{y_n}{2}\right)$