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:

\int\limits_a^b f(x)dx\approx\sum_{i=1}^nf(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[edit]

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}^nf(x_i)\Delta x

Left Rectangle[edit]

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}^nf(x_{i-1})\Delta x

Trapezoidal Rule[edit]

\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[edit]

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]

Further reading[edit]

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Integration techniques/Numerical Approximations