Chapter 6.5, Problem 13E

(a)

To determine

To calculate: The value of the integral using Simpson’s rule with n=4

Answer to Problem 13E

The value of the integral is ${\displaystyle \underset{0}{\overset{2}{\int }}xdx}=2$

Explanation of Solution

Given information:

Using Simpson’s rule with n=4 approximation, the given integral is

  ${\displaystyle {\int }_0^{2}xdx}$

Formula used:

  $S=\frac{h}{3}\left(y_0+4y_1+2y_2+4y_3+\mathrm{.....}+2y_{n-2}+4y_{n-1}+y_n\right)$

Calculation:

To approximate ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$

  $S=\frac{h}{3}\left(y_0+4y_1+2y_2+4y_3+\mathrm{.....}+2y_{n-2}+4y_{n-1}+y_n\right)$

Where [a, b] is partitioned into an even number n of subintervals of equal length $h=\frac{b-a}{n}$

Now, apply the Simpson’s rule with

  $\begin{array}{l}n=4\\ f\left(x\right)=x\\ [a,b]=[0,2]\end{array}$

So,

  $\begin{array}{l}h=\frac{b-a}{n}\\ =\frac{2-0}{4}\\ h=\frac{1}{2}\end{array}$

  

Now, substitute the values from given table in this equation

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{2}{\int }}xdx=\frac{h}{3}\left(y_0+4y_1+2y_2+4y_3+y_4\right)}\\ =\frac{\frac{1}{2}}{3}\left(0+4\left(\frac{1}{2}\right)+2\left(1\right)+4\left(\frac{3}{4}\right)+2\right)\\ =\frac{1}{3}\left(0+2+2+6+2\right)\\ {\displaystyle \underset{0}{\overset{2}{\int }}xdx}=2\end{array}$

Conclusion:

The value of the integral is ${\displaystyle \underset{0}{\overset{2}{\int }}xdx}=2$

(b)

To determine

To calculate:The exact value of the integral is ${\displaystyle {\int }_0^{2}xdx}$

Answer to Problem 13E

The value of the integral is ${\displaystyle \underset{0}{\overset{2}{\int }}xdx=2}$

Explanation of Solution

Given information:

Using Simpson’s rule with n=4 approximation, the given integral is

  ${\displaystyle {\int }_0^{2}xdx}$

Formula used:

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=F\left(b\right)-F\left(a\right)$

Calculation:

If f is continuous at every point of [a, b] , and if F is any anti-derivative of f on [a, b] , then

  ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx=F\left(b\right)-F\left(a\right)$

This part of the fundamental theorem is also called the integral evaluation theorem.

Now, use the above formula to find the value of integral.

  $f\left(x\right)=x$

If F is any anti-derivative of f then

  $F\left(x\right)=\frac{x^2}{2}$

So,

  $\begin{array}{l}{\displaystyle \underset{0}{\overset{2}{\int }}xdx=\left(\frac{2^2}{2}-\frac{0^2}{2}\right)}\\ =\frac{4}{2}-0\\ {\displaystyle \underset{0}{\overset{2}{\int }}xdx=2}\end{array}$

Conclusion:

The value of the integral is ${\displaystyle \underset{0}{\overset{2}{\int }}xdx=2}$