# Calculus/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

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

where is any point in the i-th sub-interval on .

### Right Rectangle

[edit | edit source]A special case of the Riemann sum, where we let , in other words the point on the far right-side of each sub-interval on, . Again if we pick n to be finite, then we have:

### Left Rectangle

[edit | edit source]Another special case of the Riemann sum, this time we let , which is the point on the far left side of each sub-interval on . As always, this is an approximation when is finite. Thus, we have:

## Trapezoidal Rule

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## Simpson's Rule

[edit | edit source]Remember, n must be even,

## Maclaurin Approximation

[edit | edit source]A common technique of approximating common trigonometric functions is to use the Taylor-Maclaurin series. Term-by-term integration allows one to easy compute the value of the integral by hand, well up to 5 decimal places of precision, and up to 10 given a factorial table.

For example, using the Maclaurin series of , one can easily approximate its integral with a polynomial.

We can then easily integrate each term, taking and to be constants.

We can easily find the constant term by inspecting the known principle integral, , and the new series. This nets us the final equation.

While this is a rather fast-converging series, converging at digits of significance, it is relatively useless, since factorials are expensive to compute.