Chapter 5.5, Problem 1E

Approximate the integrals, using $n=2$ Gaussian Quadrature. Compare with the correct value, and give the approximation error.

1. ${\displaystyle {\int }_{-1}^1\left(x^3+2x\right)dx}$
2. ${\displaystyle {\int }_{-1}^1{x}^{4}dx}$
3. ${\displaystyle {\int }_{-1}^1{e}^{x}dx}$
4. ${\displaystyle {\int }_{-1}^1\cos \pi xdx}$

To determine

To find out approximation error to solve the integral using $n=2$ Gaussian quadrature.

Answer to Problem 1E

  $\begin{array}{l}\text{Correct value }=0;\\ \text{error}=0\end{array}$

Explanation of Solution

Given information:

The given expression is ${\displaystyle \underset{-1}{\overset{1}{\int }}(x^3+2x)}dx$ .and solve for $n=2$ Gaussian quadrature

Calculation:

It’s known from the given $n=2$ Gaussian quadrature coefficient are $\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}}$ .

And given expression is ${\displaystyle \underset{-1}{\overset{1}{\int }}(x^3+2x)}dx$

Using $n=2$ Gaussian quadrature by use of the approximate integral

$\begin{array}{l}f(x)=x^3+2x\\ \text{for considering both coefficients }\\ f(\frac{1}{\sqrt{3}})=x^3+2x=\frac{1}{3\sqrt{3}}+\frac{2}{\sqrt{3}}\\ f(-\frac{1}{\sqrt{3}})=x^3+2x=\frac{-1}{3\sqrt{3}}+\frac{-2}{\sqrt{3}}\\ f(\frac{1}{\sqrt{3}})+f(-\frac{1}{\sqrt{3}})=0\left\{\text{odd power}\right\}\end{array}$

 Hence, $f(\frac{-1}{\surd 3})+f(\frac{1}{\surd 3})=0$ , which agrees with the correct value of the integral. Correct value is error   $\text{=0}$

To determine

To find out approximation error to solve the integral using $n=2$ Gaussian quadrature.

Answer to Problem 1E

Correct value is  $\text{0}\text{.2222},\text{ and the error} =\frac{8}{45}\approx 0.177778$ .

Explanation of Solution

Given information:

The given expression is ${\displaystyle \underset{-1}{\overset{1}{\int }}{x}^4}dx$ .and solve for $n=2$ Gaussian quadrature Calculation:

It’s known from the given $n=2$ Gaussian quadrature coefficient are $\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}}$ .

The given expression is ${\displaystyle \underset{-1}{\overset{1}{\int }}{x}^4}dx$ .

Using $n=2$ Gaussian quadrature by use of the approximate integral

$\begin{array}{l}f(x)=x^4\\ \text{for considering both coefficients }\\ f(\frac{1}{\sqrt{3}})=x^4=\frac{1}{9}\\ f(-\frac{1}{\sqrt{3}})=x^4=\frac{-1}{9}\\ f(\frac{-1}{\surd 3})+f(\frac{1}{\surd 3})=\frac{1}{9}+\frac{1}{9}=\frac{2}{9}\text{ =0}\text{.2222 }\left\{\text{even power}\right\}\end{array}$

 Hence, $f(\frac{-1}{\surd 3})+f(\frac{1}{\surd 3})=\frac{2}{9}$ , which agrees with the correct value of the integral. Correct value is  $\text{0}\text{.2222},\text{ and the error} =\frac{8}{45}\approx 0.177778$ .

To determine

To find out approximation error to solve the integral using $n=2$ Gaussian quadrature.

Answer to Problem 1E

Correct value $2.342696$ , the error $=0.007706$ .

Explanation of Solution

Given information:

The given expression is ${\displaystyle \underset{-1}{\overset{1}{\int }}{e}^x}dx$ .and solve for $n=2$ Gaussian quadrature Calculation:

Its known from the given $n=2$ Gaussian quadrature coefficient are $\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}}$ .

The given expression is ${\displaystyle \underset{-1}{\overset{1}{\int }}{e}^x}dx$ .

Using $n=2$ by use of the integral formula   $f(\frac{-1}{\surd 3})+f(\frac{1}{\surd 3})=e^{\frac{-1}{\surd 3}}+e^{\frac{1}{\surd 3}}\approx 2.342696$ .

Compared to the correct value $2.342696$ , the error  $=0.007706$ .

To determine

To find out approximation error to solve the integral using $n=2$ Gaussian quadrature.

Answer to Problem 1E

Correct value $-0.481237$ , the error $=0.481237$ .

Explanation of Solution

Given information:

The given expression is ${\displaystyle \underset{-1}{\overset{1}{\int }}\cos \pi x}dx$ .and solve for n=2

Calculation:

Its known from the given $n=2$ Gaussian quadrature coefficient are $\frac{1}{\sqrt{3}},\frac{-1}{\sqrt{3}}$ .

The given expression is ${\displaystyle \underset{-1}{\overset{1}{\int }}\cos \pi x}dx$ .

Using $n=2$ by use of the integral formula

  $f(-1\surd 3)+f(1\surd 3)=cos\frac{-\pi }{\sqrt{3}}+cos\frac{\pi }{\sqrt{3}}\approx -0.481237$ .

Compared to the correct value $-0.481237$ , the error $=0.481237$ .