Gaussian Quadrature - Change of Interval

Change of Interval

An integral over must be changed into an integral over before applying the Gaussian quadrature rule. This change of interval can be done in the following way:

$\int_a^b f(x)\,dx = \frac{b-a}{2} \int_{-1}^1 f\left(\frac{b-a}{2}z + \frac{a+b}{2}\right)\,dz.$

After applying the Gaussian quadrature rule, the following approximation is:

$\int_a^b f(x)\,dx \approx \frac{b-a}{2} \sum_{i=1}^n w_i f\left(\frac{b-a}{2}z_i + \frac{a+b}{2}\right).$

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