# Calculus/Integration techniques/Numerical Approximations

 ← Integration techniques/Irrational Functions Calculus Integration/Exercises → Integration techniques/Numerical Approximations

It is often the case, when evaluating definite integrals, that an antiderivative for the integrand cannot be found, or is extremely difficult to find. In some instances, a numerical approximation to the value of the definite value will suffice. The following techniques can be used, and are listed in rough order of ascending complexity.

## Riemann Sum

This comes from the definition of an integral. If we pick n to be finite, then we have:

$\int \limits _{a}^{b}f(x)dx\approx \sum _{i=1}^{n}f(x_{i}^{*})\Delta x$

where $x_{i}^{*}$  is any point in the i-th sub-interval $[x_{i-1},x_{i}]$  on $[a,b]$  .

### Right Rectangle

A special case of the Riemann sum, where we let $x_{i}^{*}=x_{i}$  , in other words the point on the far right-side of each sub-interval on, $[a,b]$  . Again if we pick n to be finite, then we have:

$\int \limits _{a}^{b}f(x)dx\approx \sum _{i=1}^{n}f(x_{i})\Delta x$

### Left Rectangle

Another special case of the Riemann sum, this time we let $x_{i}^{*}=x_{i-1}$  , which is the point on the far left side of each sub-interval on $[a,b]$  . As always, this is an approximation when $n$  is finite. Thus, we have:

$\int \limits _{a}^{b}f(x)dx\approx \sum _{i=1}^{n}f(x_{i-1})\Delta x$

## Trapezoidal Rule

$\int \limits _{a}^{b}f(x)dx\approx {\frac {b-a}{2n}}\left[f(x_{0})+2\sum _{i=1}^{n-1}{\bigl (}f(x_{i}){\bigr )}+f(x_{n})\right]={\frac {b-a}{2n}}{\bigg (}{f(x_{0})+2f(x_{1})+2f(x_{2})+\cdots +2f(x_{n-1})+f(x_{n})}{\bigg )}$

## Simpson's Rule

Remember, n must be even,

 $\int \limits _{a}^{b}f(x)dx$ $\approx {\frac {b-a}{6n}}\left[f(x_{0})+\sum _{i=1}^{n-1}\left((3-(-1)^{i})f(x_{i})\right)+f(x_{n})\right]$ $={\frac {b-a}{6n}}{\bigg [}f(x_{0})+4f{\bigl (}{\tfrac {x_{1}}{2}}{\bigr )}+2f(x_{1})+4f{\bigl (}{\tfrac {x_{3}}{2}}{\bigr )}+\cdots +4f{\bigl (}{\tfrac {x_{n-1}}{2}}{\bigr )}+f(x_{n}){\bigg ]}$ 