Chapter 6.5, Problem 28E

(a)

To determine

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

Answer to Problem 28E

The value of integral is 2.00

Explanation of Solution

Given information:

The value of function is ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$

Formula used:

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

Calculation:

In Simpson’s rule, to approximate ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$ use

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

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

Now,

To find integral use your calculator program of Simpson's rule approximations with

  $\begin{array}{l}f\left(x\right)=\sin x\\ \left[a,b\right]=\left[0,\pi \right]\\ n=10,100\text{and}1000\end{array}$

  

Conclusion:

The value of integral is 2.00

(b)

To determine

To calculate:The exact value of the integral is ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$

Answer to Problem 28E

The value of the integral is 0 when n=1000

Explanation of Solution

Given information:

The value of function is ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$

Formula used:

  $\left|E_S\right|\le \frac{b-a}{180}{h}^4{M}_{f^{\left(4\right)}}$

Calculation:

If S represents the approximation to ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$ given by the Simpson's rule, then the errors E_S satisfy

  $\left|E_S\right|\le \frac{b-a}{180}{h}^4{M}_{f^{\left(4\right)}}$

  $M_{f^{\left(4\right)}}$ , It is maximum value of $\left|f^{\left(4\right)}\right|\text{on}\left[a,b\right]$

Here, use your calculator to find the error, error is also given by

  $\begin{array}{l}\text{error}\left(\left|E_S\right|\right)={\displaystyle \underset{0}{\overset{\pi }{\int }}\sin x-T_n}\\ ={\left[-\cos x\right]}_0^x-T\\ =-\cos \pi -\left[-\cos 0\right]-T_n\\ \text{error}\left(\left|E_S\right|\right)=2-T_n\end{array}$

Where T_n represents the value of integral by trapezoidal rule for different value or n

  

Conclusion:

The value of the integral is 0 when n =1000

(c)

To determine

To calculate:the value of error.

Answer to Problem 28E

  $\left|E_{S_{10n}}\right|\approx {10}^{-4}\left|E_{Tn}\right|$

Explanation of Solution

Given information:

The value of function is ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$

Formula used:

The error gets multiplied

Calculation:

Observe the last column or above table, error gets multiplied approximately by 10-4

So,

  $\left|E_{S_{10n}}\right|\approx {10}^{-4}\left|E_{Tn}\right|$

Conclusion:

  $\left|E_{S_{10n}}\right|\approx {10}^{-4}\left|E_{Tn}\right|$

(d)

To determine

To find:The value of error bound.

Answer to Problem 28E

  $\left|E_{S_{10n}}\right|\le \left|E_{S_n}\right|\times {10}^{-4}$

Explanation of Solution

Given information:

The value of function is ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)}dx$

Formula used:

  $\left|E_S\right|\le \frac{b-a}{180}{h}^4{M}_{f^{\left(4\right)}}$

Calculation:

Now, use the formula for error bound

  $\left|E_S\right|\le \frac{b-a}{180}{h}^4{M}_{f^{\left(4\right)}}$

Now

  $h=\frac{b-a}{n}$

So,

  $\begin{array}{l}\left|E_{S_n}\right|\le \frac{\pi -0}{180}\times {\left(\frac{\pi -0}{n}\right)}^4{M}_{f^{\left(4\right)}}\\ \le \frac{{\pi }^5}{180n^4}\times M_{f^{\left(4\right)}}\\ \left|E_{S_{10n}}\right|\le \frac{\pi -0}{180}\times {\left(\frac{\pi -0}{10n}\right)}^4{M}_{f^{\left(4\right)}}\\ \le \frac{{\pi }^5}{180n^4}\times {10}^{-4}\times M_{f^{\left(4\right)}}\\ \le \left|E_{S_n}\right|\times {10}^{-4}\end{array}$

Hence, error bound accounts for the pattern.

Conclusion:

The error bound gets account for the pattern.