MHB How to Use Gauss Chebyshev Formula for Approximating Integrals with n=3?

[jiasyuen]

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral.

$$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1.

Also compare the result with true value, where the zeros and the corresponding weights of the following simple set of orthogonal polynomial is given as below.

$x_i=0,\frac{\sqrt{3}}{2},-\frac{\sqrt{3}}{2}$

$w_i=1,2,2$

$i=1,2,3$I visited the wikipedia about this formula, but I really don't know how to use it. Can anyone guide me?

 

[chisigma]

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral.

$$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1.

Also compare the result with true value, where the zeros and the corresponding weights of the following simple set of orthogonal polynomial is given as below.

$x_i=0,\frac{\sqrt{3}}{2},-\frac{\sqrt{3}}{2}$

$w_i=1,2,2$

$i=1,2,3$I visited the wikipedia about this formula, but I really don't know how to use it. Can anyone guide me?

The Gauss Chebyshef of order n approximates the integral as...

$\displaystyle \int_{-1}^{1} \frac{f(x)}{\sqrt{1 - x^{2}}}\ dx \sim \sum_{i=1}^{n} w_{i}\ f(x_{i})\ (1)$

In Your case is...

$n=3$

$ f(x)=x^{4}$

$x_i=0,\frac{\sqrt{3}}{2},-\frac{\sqrt{3}}{2}$

$w_i=1,2,2$

Kind regards

$\chi$ $\sigma$

 

[chisigma]

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral.

$$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1.

Also compare the result with true value, where the zeros and the corresponding weights of the following simple set of orthogonal polynomial is given as below.

$x_i=0,\frac{\sqrt{3}}{2},-\frac{\sqrt{3}}{2}$

$w_i=1,2,2$

$i=1,2,3$I visited the wikipedia about this formula, but I really don't know how to use it. Can anyone guide me?

Probably there are some mistakes in the $x_{i}$ and $w_{i}$ because is...

$\displaystyle x_{i} = \cos (\frac{2\ i -1}{2\ n}\ \pi), w_{i} = \frac{\pi}{n}\ (1)$

... and for n=3... $\displaystyle x_{1} = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$

$\displaystyle x_{2} = \cos \frac{\pi}{2} = 0$

$\displaystyle x_{3} = \cos (\frac{5}{6}\ \pi) = - \frac{\sqrt{3}}{2}$

$\displaystyle w_{1}=w_{2}=w_{3}= \frac{\pi}{6}$

... so that applying the Gauss Chebyshef formula You obtain...

$\displaystyle \int_{-1}^{1} \frac{x^{4}}{\sqrt{1 - x^{2}}}\ d x = \frac{3}{8}\ \pi\ (2)$

... which is the exact value of the integral [a great advantage of this type of integration formula...] (Happy) ...

Kind regards

$\chi$ $\sigma$