(Solved): The weights and nodes for the three point Gaussian quadrature (GQ) formula for integrals of the ty ...

The weights and nodes for the three point Gaussian quadrature (GQ) formula for integrals of the type \[ I(f)=\int_{-1}^{1} w(x) f(x) d x, \] where \( w(x)=\sqrt{1-x^{2}} \), are given by \( w_{0}=\pi / 8, w_{1}=\pi / 4, w_{2}=\pi / 8 \), and \( x_{0}=-\sqrt{2} / 2, x_{1}=0 \), \( x_{2}=\sqrt{2} / 2 \). From these nodes and weights, what is the value of \( I\left(x^{4}\right) ? \) a. \( \frac{\pi \sqrt{2}}{32} \) c. 0 b. \( \frac{\pi}{16} \) d. \( I\left(x^{4}\right) \) cannot be determined exactl