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[edit]

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]

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]

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[edit]

Simpson's Rule[edit]

Remember, n must be even,

Further reading[edit]

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Integration techniques/Numerical Approximations