Calculus/Integration techniques/Numerical Approximations

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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

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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

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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

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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

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Remember, n must be even,

   
 

Maclaurin Approximation

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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.

Further reading

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Integration techniques/Numerical Approximations