Question

6. So, does it work? Does this actually lead to a method that is good for functions in general? Use 2-point Gaussian quadrature to approximate the following integral esin(x) daz The actual value of this integral rounded to 4 decimal places is -0.6635

Answer 1

$a=-1,b=1,f(x)=e^{-x}\sin(x)$ for
2
points and weights are $w_1=w_2=1$ and nodes are
L 02

Based on 2 point Gaussian quadrature, we get the approximate integral as

$w_1f(x_1)+w_2f(x_2)=f\left(-\frac{1}{\sqrt{3}} \right )+f\left(\frac{1}{\sqrt{3}} \right )$ where

) ~-0.97225, f

Thus, we have (rounded to 5 decimal places)

~-066584 V3

Which is quite close to the actual value given

0,6635

$\blacksquare$

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