HW2.tex

\documentclass{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=5.50.0.2960}
%TCIDATA{<META NAME="SaveForMode" CONTENT="1">}
%TCIDATA{BibliographyScheme=Manual}
%TCIDATA{Created=Saturday, December 03, 2022 21:48:08}
%TCIDATA{LastRevised=Thursday, December 08, 2022 20:40:32}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="DocumentShell" CONTENT="Standard LaTeX\Blank - Standard LaTeX Article">}
%TCIDATA{CSTFile=40 LaTeX article.cst}
%TCIDATA{ComputeGeneralSettings=0,10,5,0,0,0,0}
%TCIDATA{ComputeDefs=
%$b=6$
%$a=-8.2$
%$q(x)=-9.\,\allowbreak 502\,1\times 10^{-7}x^{7}+4.\,\allowbreak
%717\,3\times 10^{-6}x^{6}+1.\,\allowbreak 712\,9\times
%10^{-4}x^{5}+5.\,\allowbreak 213\,7\times 10^{-4}x^{4}-2.\,\allowbreak
%073\,1\times 10^{-3}x^{3}-4.\,\allowbreak 298\,6\times
%10^{-3}x^{2}+3.\,\allowbreak 018\,7\times 10^{-2}x-1.\,\allowbreak 297\,8$
%$p(x)=\bigskip -7.\,\allowbreak 634\,4\times 10^{-9}x^{9}-2.\,\allowbreak
%736\,9\times 10^{-7}x^{8}-5.\,\allowbreak 319\,9\times
%10^{-7}x^{7}+2.\,\allowbreak 444\,3\times 10^{-5}x^{6}+1.\,\allowbreak
%677\,6\times 10^{-4}x^{5}-9.\,\allowbreak 197\,7\times
%10^{-5}x^{4}-2.\,\allowbreak 736\,9\times 10^{-3}x^{3}-2.\,\allowbreak
%483\,6\times 10^{-3}x^{2}+0.030\,13x+0.493\,56$
%$\bigskip f(x)=p(x)-q(x)$
%$h=\frac{b-a}{n+2}$
%$g(j)=a+(j+1)h$
%$r(x)=\frac{1}{2}f(x)$
%$k(x)=f(x)\sqrt{4+f^{\prime }(x)^{2}}$
%$l(x)=5.\,\allowbreak 828\,4\times 10^{-17}x^{18}+4.\,\allowbreak
%178\,9\times 10^{-15}\allowbreak x^{17}+6.\,\allowbreak 852\times
%10^{-14}x^{16}-5.\,\allowbreak 301\,2\times 10^{-13}\allowbreak
%x^{15}-1.\,\allowbreak 056\,9\times 10^{-11}x^{14}+2.\,\allowbreak
%779\,7\times 10^{-11}\allowbreak x^{13}+7.\,\allowbreak 320\,3\times
%10^{-10}x^{12}-3.\,\allowbreak 166\,6\times 10^{-10}\allowbreak
%x^{11}-2.\,\allowbreak 573\,3\times 10^{-8}x^{10}-4.\,\allowbreak
%766\,1\times 10^{-8}\allowbreak x^{9}-5.\,\allowbreak 281\,3\times
%10^{-7}x^{8}+2.\,\allowbreak 297\,6\times 10^{-6}\allowbreak
%x^{7}+6.\,\allowbreak 888\,9\times 10^{-5}x^{6}-1.\,\allowbreak 498\,7\times
%10^{-5}\allowbreak x^{5}-2.\,\allowbreak 194\,1\times
%10^{-3}x^{4}-2.\,\allowbreak 378\,5\times 10^{-3}\allowbreak
%x^{3}+6.\,\allowbreak 502\,8\times 10^{-3}x^{2}-2.\,\allowbreak 042\,2\times
%10^{-4}\allowbreak x+3.\,\allowbreak 209\,1$
%$S(n)=\frac{4}{3}h\sum\limits_{j=0}^{n/4}(2f(g(4j))-f(g(4j+1))+2f(g(4j+2)))$
%$n=160$
%}


\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}
\begin{document}


\subsection{$p(x)=-7.\,\allowbreak 634\,4\times 10^{-9}x^{9}-2.\,\allowbreak
736\,9\times 10^{-7}x^{8}-5.\,\allowbreak 319\,9\times
10^{-7}x^{7}+2.\,\allowbreak 444\,3\times 10^{-5}x^{6}+1.\,\allowbreak
677\,6\times 10^{-4}x^{5}-9.\,\allowbreak 197\,7\times
10^{-5}x^{4}-2.\,\allowbreak 736\,9\times 10^{-3}x^{3}-2.\,\allowbreak
483\,6\times 10^{-3}x^{2}+0.030\,13x+0.493\,56\allowbreak $\protect\FRAME{%
dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific
Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display
"USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin
"-9";xmax "9";xviewmin "-9";xviewmax "9";yviewmin "-5";yviewmax
"5";viewset"XY";rangeset"X";plottype 4;axesFont "Times New
Roman,12,0000000000,useDefault,normal";numpoints 100;plotstyle
"patch";axesstyle "normal";axestips FALSE;xis \TEXUX{x};var1name
\TEXUX{$x$};function \TEXUX{$-7.\,\allowbreak 634\,4\times
10^{-9}x^{9}-2.\,\allowbreak 736\,9\times 10^{-7}x^{8}-5.\,\allowbreak
319\,9\times 10^{-7}x^{7}+2.\,\allowbreak 444\,3\times
10^{-5}x^{6}+1.\,\allowbreak 677\,6\times 10^{-4}x^{5}-9.\,\allowbreak
197\,7\times 10^{-5}x^{4}-2.\,\allowbreak 736\,9\times
10^{-3}x^{3}-2.\,\allowbreak 483\,6\times
10^{-3}x^{2}+0.030\,13x+0.493\,56$};linecolor "black";linestyle 1;pointstyle
"point";linethickness 1;lineAttributes "Solid";var1range
"-9,9";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle
"Line";function \TEXUX{$-9.\,\allowbreak 502\,1\times
10^{-7}x^{7}+4.\,\allowbreak 717\,3\times 10^{-6}x^{6}+1.\,\allowbreak
712\,9\times 10^{-4}x^{5}+5.\,\allowbreak 213\,7\times
10^{-4}x^{4}-2.\,\allowbreak 073\,1\times 10^{-3}x^{3}-4.\,\allowbreak
298\,6\times 10^{-3}x^{2}+3.\,\allowbreak 018\,7\times
10^{-2}x-1.\,\allowbreak 297\,8\allowbreak $};linecolor "black";linestyle
1;pointstyle "point";linethickness 1;lineAttributes "Solid";var1range
"-9,9";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle
"Line";VCamFile 'RMKNGJ0K.xvz';valid_file "T";tempfilename
'RMKNFP07.wmf';tempfile-properties "XPR";}}}

\subsection{$q(x)=-9.\,\allowbreak 502\,1\times 10^{-7}x^{7}+4.\,\allowbreak
717\,3\times 10^{-6}x^{6}+1.\,\allowbreak 712\,9\times
10^{-4}x^{5}+5.\,\allowbreak 213\,7\times 10^{-4}x^{4}-2.\,\allowbreak
073\,1\times 10^{-3}x^{3}-4.\,\allowbreak 298\,6\times
10^{-3}x^{2}+3.\,\allowbreak 018\,7\times 10^{-2}x-1.\,\allowbreak
297\,8\allowbreak $}

\subsection{$\protect\bigskip f(x)=p(x)-q(x)=\allowbreak -7.\,\allowbreak
634\,4\times 10^{-9}x^{9}-2.\,\allowbreak 736\,9\times
10^{-7}x^{8}+4.\,\allowbreak 182\,2\times 10^{-7}\allowbreak
x^{7}+1.\,\allowbreak 972\,6\times 10^{-5}x^{6}-3.\,\allowbreak 53\times
10^{-6}x^{5}-6.\,\allowbreak 133\,5\times 10^{-4}\allowbreak
x^{4}-6.\,\allowbreak 638\times 10^{-4}x^{3}+1.\,\allowbreak 815\times
10^{-3}x^{2}-5.\,\allowbreak 7\times 10^{-5}x+1.\,\allowbreak
791\,4\allowbreak $}

\subsection{$f^{(4)}(x)=\allowbreak -2.\,\allowbreak 308\,6\times
10^{-5}x^{5}-4.\,\allowbreak 598\,0\times 10^{-4}x^{4}+\allowbreak
3.\,\allowbreak 513\times 10^{-4}x^{3}+7.\,\allowbreak 101\,3\times
10^{-3}x^{2}-4.\,\allowbreak 236\times 10^{-4}\allowbreak x-0.014\,72$%
\protect\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special{language
"Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth
0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine
"MuPAD";xmin "-5.001000";xmax "5.001000";xviewmin "-5.001000";xviewmax
"5.001000";yviewmin "-0.154929";yviewmax
"0.5";viewset"XY";rangeset"X";plottype 4;axesFont "Times New
Roman,12,0000000000,useDefault,normal";numpoints 100;plotstyle
"patch";axesstyle "normal";axestips FALSE;xis \TEXUX{x};var1name
\TEXUX{$x$};function \TEXUX{$-2.\,\allowbreak 308\,6\times
10^{-5}x^{5}-4.\,\allowbreak 598\,0\times 10^{-4}x^{4}+\allowbreak
3.\,\allowbreak 513\times 10^{-4}x^{3}+7.\,\allowbreak 101\,3\times
10^{-3}x^{2}-4.\,\allowbreak 236\times 10^{-4}\allowbreak
x-0.014\,72$};linecolor "black";linestyle 1;pointstyle "point";linethickness
1;lineAttributes "Solid";var1range "-5.001000,5.001000";num-x-gridlines
100;curveColor "[flat::RGB:0000000000]";curveStyle "Line";VCamFile
'RMKL0R00.xvz';valid_file "T";tempfilename
'RMKL0R00.wmf';tempfile-properties "XPR";}}}

\subsection{\protect\bigskip $f^{(4)}(2.\,\allowbreak 804\,1)=\allowbreak
1.\,\allowbreak 524\,4\times 10^{-2}$}

\subsection{$f^{(4)}(2.\,\allowbreak 976\,2\times 10^{-2})=\allowbreak
-1.\,\allowbreak 472\,7\times 10^{-2}$}

\subsection{$f^{(4)}(-\allowbreak 2.\,\allowbreak 744\,2)=\allowbreak
1.\,\allowbreak 017\,7\times 10^{-2}$}

\subsection{$f^{(4)}(-16.\,\allowbreak 023\allowbreak )=\allowbreak
-5.\,\allowbreak 554\,6$}

$%
\begin{array}{cc}
x & y \\ 
2.\,\allowbreak 804\,1 & 1.\,\allowbreak 524\,4\times 10^{-2} \\ 
-\allowbreak 2.\,\allowbreak 744\,2 & \allowbreak 1.\,\allowbreak
017\,7\times 10^{-2}%
\end{array}%
$

\subsection{$f^{(5)}(x)=\allowbreak -1.\,\allowbreak 154\,3\times
10^{-4}x^{4}-1.\,\allowbreak 839\,2\times 10^{-3}x^{3}+\allowbreak
1.\,\allowbreak 053\,9\times 10^{-3}x^{2}+1.\,\allowbreak 420\,3\times
10^{-2}x-4.\,\allowbreak 236\times 10^{-4}\allowbreak =0$, Solution is: $%
2.\,\allowbreak 804\,1,2.\,\allowbreak 976\,2\times 10^{-2},-\allowbreak
2.\,\allowbreak 744\,2,-16.\,\allowbreak 023\allowbreak $}

\subsection{$a=-8.2$}

\subsection{$b=6$}

\bigskip

\bigskip

==============================================================1=============================================================

\subsection{ $\protect\int_{a}^{b}(p(x)-q(x))dx=\allowbreak 23.\,\allowbreak
519\,555\,61$}

\subsection{$h=\frac{b-a}{n+2}$}

\section{$g(j)=a+(j+1)h$}

\subsection{$\protect\int_{a}^{b}(p(x)-q(x))dx\approx \frac{4}{3}%
h\sum\limits_{j=0}^{n/4}(2f(g(4j))-f(g(4j+1))+2f(g(4j+2)))$}

Theoretrical error :$%
\begin{array}{cc}
x & y \\ 
2.\,\allowbreak 804\,1 & 1.\,\allowbreak 524\,4\times 10^{-2}%
\end{array}%
$

\subsection{$\frac{7h^{4}}{90}(b-a)\times 1.\,\allowbreak 524\,4\times
10^{-2}=\allowbreak 2.\,\allowbreak 922\,2\times 10^{-3}$}

\subsection{\protect\bigskip $n=20$}

\subsection{$S(n)=\frac{4}{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))-f(g(4j+1))+2f(g(4j+2)))=\allowbreak
24.\,\allowbreak 967\,127\,68$}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}(p(x)-q(x))dx\allowbreak
\allowbreak =\allowbreak 24.\,\allowbreak 967\,127\,68-\allowbreak
23.\,\allowbreak 519\,555\,61$ = $1.\,\allowbreak 447\,572\,07$}

\subsection{$n=40$}

\subsection{$S(n)=\frac{4}{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))-f(g(4j+1))+2f(g(4j+2)))=\allowbreak
24.\,\allowbreak 382\,392\,82$}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}(p(x)-q(x))dx\allowbreak
=24.\,\allowbreak 382\,392\,82-\allowbreak 23.\,\allowbreak
519\,555\,61=\allowbreak 0.862\,837\,21$ }

\subsection{$n=60$}

\subsection{$S(n)=\frac{4}{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))-f(g(4j+1))+2f(g(4j+2)))=\allowbreak
24.\,\allowbreak 123\,553\,79$}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}(p(x)-q(x))dx\allowbreak
\allowbreak =24.\,\allowbreak 123\,553\,79-\allowbreak 23.\,\allowbreak
519\,555\,61=\allowbreak 0.603\,998\,18$}

\bigskip

\subsection{$n=80$}

\subsection{$S(n)=\frac{4}{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))-f(g(4j+1))+2f(g(4j+2)))=\allowbreak
23.\,\allowbreak 983\,063\,82$}

\subsection{\protect\bigskip $E(n)=S(n)-\protect\int_{a}^{b}(p(x)-q(x))dx%
\allowbreak \allowbreak =\allowbreak 23.\,\allowbreak
983\,063\,82-23.\,\allowbreak 519\,555\,61=\allowbreak 0.463\,508\,21$}

\bigskip

\subsection{$n=160$}

\subsection{$S(n)=\frac{4}{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))-f(g(4j+1))+2f(g(4j+2)))=\allowbreak
23.\,\allowbreak 759\,020\,61$}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}(p(x)-q(x))dx\allowbreak
\allowbreak =\allowbreak 23.\,\allowbreak 759\,020\,61-23.\,\allowbreak
519\,555\,61=\allowbreak 0.239\,465\,$}

\bigskip

\subsection{$n=10000000$}

\subsection{$S(n)=\frac{4}{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))-f(g(4j+1))+2f(g(4j+2)))=\allowbreak
23.\,\allowbreak 559$}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}(p(x)-q(x))dx\allowbreak
\allowbreak =\allowbreak 23.\,\allowbreak 559-23.\,\allowbreak 520=0.039\,$}

\bigskip

\subsection{\protect\bigskip $%
\protect\begin{array}{ccc}
n & S(n) & E(n)\protect \\ 
20 & 24.\,\allowbreak 967 & 1.\,\allowbreak 447\,572\,07\protect \\ 
40 & 24.\,\allowbreak 382 & \allowbreak 0.862\,837\,21\protect \\ 
60 & 24.\,\allowbreak 124 & \allowbreak 0.603\,998\,18\protect \\ 
80 & 23.\,\allowbreak 983 & \allowbreak 0.463\,508\,21\protect \\ 
160 & 23.\,\allowbreak 759 & \allowbreak 0.239\,465\,%
\protect\end{array}%
$}

\bigskip

\bigskip $x=\ln (%
\begin{array}{c}
20 \\ 
40 \\ 
60 \\ 
80 \\ 
160%
\end{array}%
)=\allowbreak x=%
\begin{array}{c}
2.\,\allowbreak 995\,732\,274 \\ 
3.\,\allowbreak 688\,879\,454 \\ 
4.\,\allowbreak 094\,344\,562 \\ 
4.\,\allowbreak 382\,026\,635 \\ 
5.\,\allowbreak 075\,173\,815%
\end{array}%
\allowbreak $

$y=\ln (%
\begin{array}{c}
1.\,\allowbreak 447\,572\,07 \\ 
\allowbreak 0.862\,837\,21 \\ 
\allowbreak 0.603\,998\,18 \\ 
\allowbreak 0.463\,508\,21 \\ 
\allowbreak 0.239\,465\,%
\end{array}%
)=\allowbreak y=%
\begin{array}{c}
0.369\,887\,718\,\allowbreak 5 \\ 
-0.147\,529\,238\,\allowbreak 4 \\ 
-0.504\,184\,094\,\allowbreak 3 \\ 
-0.768\,931\,181\,\allowbreak 1 \\ 
-1.\,\allowbreak 429\,348\,011%
\end{array}%
\allowbreak $

$%
\begin{array}{cc}
x & y \\ 
2.\,\allowbreak 995\,732\,274 & 0.369\,887\,718\,\allowbreak 5 \\ 
3.\,\allowbreak 688\,879\,454 & -0.147\,529\,238\,\allowbreak 4 \\ 
4.\,\allowbreak 094\,344\,562 & -0.504\,184\,094\,\allowbreak 3 \\ 
4.\,\allowbreak 382\,026\,635 & -0.768\,931\,181\,\allowbreak 1 \\ 
5.\,\allowbreak 075\,173\,815 & -1.\,\allowbreak 429\,348\,011%
\end{array}%
$, Polynomial fit: $y=3.\,\allowbreak 017\,0-0.868\,23\allowbreak x$\FRAME{%
dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special{language "Scientific
Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth 0pt;display
"USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine "MuPAD";xmin
"2.5";xmax "5.5";xviewmin "2.5";xviewmax "5.5";yviewmin "-2";yviewmax
"0";viewset"XY";rangeset"X";plottype 4;axesFont "Times New
Roman,12,0000000000,useDefault,normal";numpoints 100;plotstyle
"patchnogrid";axesstyle "normal";axestips FALSE;xis \TEXUX{x};var1name
\TEXUX{$x$};function
\TEXUX{$\MATRIX{2,5}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,,,,,}\CELL{2.\,%
\allowbreak 995\,7}\CELL{0.369\,49}\CELL{3.\,\allowbreak
688\,9}\CELL{-0.148\,5}\CELL{4.\,\allowbreak
094\,3}\CELL{-0.504\,18}\CELL{4.\,\allowbreak
382}\CELL{-0.770\,03}\CELL{5.\,\allowbreak 075\,2}\CELL{-1.\,\allowbreak
431\,3}$};linecolor "black";linestyle 1;pointplot TRUE;pointstyle
"circle";linethickness 1;lineAttributes "Solid";curveColor
"[flat::RGB:0000000000]";curveStyle "Point";function \TEXUX{$3.\,\allowbreak
017\,0-0.868\,23\allowbreak x$};linecolor "black";linestyle 1;pointstyle
"point";linethickness 1;lineAttributes "Solid";var1range
"2.5,5.5";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle
"Line";VCamFile 'RMKM530B.xvz';valid_file "T";tempfilename
'RMKM3305.wmf';tempfile-properties "XPR";}}

\bigskip $\ln (|E|)=3.\,\allowbreak 017\,0-0.868\,23\allowbreak \ln
(n)=\allowbreak 
\begin{array}{c}
3.\,\allowbreak 017-0.868\,23\ln 20 \\ 
3.\,\allowbreak 017\,0-0.868\,23\ln 40 \\ 
3.\,\allowbreak 017\,0-0.868\,23\ln 60 \\ 
3.\,\allowbreak 017\,0-0.868\,23\ln 80 \\ 
3.\,\allowbreak 017\,0-0.868\,23\ln 160%
\end{array}%
\allowbreak $ : $%
\begin{array}{c}
0.416\,02 \\ 
-0.185\,80 \\ 
-0.537\,83 \\ 
-0.787\,61 \\ 
-1.\,\allowbreak 389\,4%
\end{array}%
\allowbreak $

\subsection{$\ln (|E|)=3.\,\allowbreak 017\,0-0.868\,23\allowbreak \ln (n)$}

\subsection{$\exp (\allowbreak 3.\,\allowbreak 017-0.868\,23\ln
n)=\allowbreak \exp \left( 3.\,\allowbreak 017-0.868\,23\ln n\right)
=\allowbreak \frac{20.\,\allowbreak 430}{n^{0.868\,23}}$}

\bigskip 

==============================================================2=============================================================

\subsection{$\protect\int_{a}^{b}\protect\pi r(x)^{2}dx=\frac{\protect\pi }{4%
}\protect\int_{a}^{b}f(x)^{2}dx=\allowbreak 9.\,\allowbreak 961\protect\pi $}

\subsection{$\frac{\protect\pi }{4}\protect\int_{a}^{b}f(x)^{2}dx\approx 
\frac{\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))^{2}-f(g(4j+1))^{2}+2f(g(4j+2))^{2})$}

\bigskip Theoretrical error :$%
\begin{array}{cc}
x & y \\ 
2.\,\allowbreak 804\,1 & 1.\,\allowbreak 524\,4\times 10^{-2}%
\end{array}%
$

\subsection{$\frac{\protect\pi }{4}\frac{7h^{4}}{90}(b-a)\times
3.\,\allowbreak 210\,6=\allowbreak 0.153\,86\protect\pi $}

\subsection{$f(x)^{2}=\allowbreak \left( 7.\,\allowbreak 634\,4\times
10^{-9}x^{9}+2.\,\allowbreak 736\,9\times 10^{-7}x^{8}-4.\,\allowbreak
182\,2\times 10^{-7}x^{7}-1.\,\allowbreak 972\,6\times
10^{-5}x^{6}+3.\,\allowbreak 53\times 10^{-6}x^{5}+6.\,\allowbreak
133\,5\times 10^{-4}x^{4}+6.\,\allowbreak 638\times
10^{-4}x^{3}-1.\,\allowbreak 815\times 10^{-3}x^{2}+5.\,\allowbreak 7\times
10^{-5}x-1.\,\allowbreak 791\,4\right) ^{2}\allowbreak $ :}

\subsection{$l(x)=5.\,\allowbreak 828\,4\times
10^{-17}x^{18}+4.\,\allowbreak 178\,9\times 10^{-15}\allowbreak
x^{17}+6.\,\allowbreak 852\times 10^{-14}x^{16}-5.\,\allowbreak 301\,2\times
10^{-13}\allowbreak x^{15}-1.\,\allowbreak 056\,9\times
10^{-11}x^{14}+2.\,\allowbreak 779\,7\times 10^{-11}\allowbreak
x^{13}+7.\,\allowbreak 320\,3\times 10^{-10}x^{12}-3.\,\allowbreak
166\,6\times 10^{-10}\allowbreak x^{11}-2.\,\allowbreak 573\,3\times
10^{-8}x^{10}-4.\,\allowbreak 766\,1\times 10^{-8}\allowbreak
x^{9}-5.\,\allowbreak 281\,3\times 10^{-7}x^{8}+2.\,\allowbreak 297\,6\times
10^{-6}\allowbreak x^{7}+6.\,\allowbreak 888\,9\times
10^{-5}x^{6}-1.\,\allowbreak 498\,7\times 10^{-5}\allowbreak
x^{5}-2.\,\allowbreak 194\,1\times 10^{-3}x^{4}-2.\,\allowbreak 378\,5\times
10^{-3}\allowbreak x^{3}+6.\,\allowbreak 502\,8\times
10^{-3}x^{2}-2.\,\allowbreak 042\,2\times 10^{-4}\allowbreak
x+3.\,\allowbreak 209\,1$}

\subsection{$l^{(4)}(x)=\allowbreak 4.\,\allowbreak 280\,4\times
10^{-12}x^{14}+2.\,\allowbreak 387\,0\times 10^{-10}\allowbreak
x^{13}+2.\,\allowbreak 993\,0\times 10^{-9}x^{12}-1.\,\allowbreak
736\,7\times 10^{-8}\allowbreak x^{11}-2.\,\allowbreak 539\,1\times
10^{-7}x^{10}+4.\,\allowbreak 770\,0\times 10^{-7}\allowbreak
x^{9}+8.\,\allowbreak 696\,5\times 10^{-6}x^{8}-2.\,\allowbreak 507\,9\times
10^{-6}x^{7}-\allowbreak 1.\,\allowbreak 296\,9\times
10^{-4}x^{6}-1.\,\allowbreak 441\,3\times 10^{-4}x^{5}-\allowbreak
8.\,\allowbreak 872\,6\times 10^{-4}x^{4}+1.\,\allowbreak 930\,0\times
10^{-3}x^{3}+0.024\,8\allowbreak x^{2}-1.\,\allowbreak 798\,4\times
10^{-3}x-5.\,\allowbreak 265\,8\times 10^{-2}$}

\subsection{$l^{(5)}(x)=\allowbreak 5.\,\allowbreak 992\,5\times
10^{-11}x^{13}+3.\,\allowbreak 103\,1\times 10^{-9}\allowbreak
x^{12}+3.\,\allowbreak 591\,5\times 10^{-8}x^{11}-1.\,\allowbreak
910\,3\times 10^{-7}\allowbreak x^{10}-2.\,\allowbreak 539\,1\times
10^{-6}x^{9}+4.\,\allowbreak 293\,0\times 10^{-6}\allowbreak
x^{8}+6.\,\allowbreak 957\,2\times 10^{-5}x^{7}-1.\,\allowbreak 755\,6\times
10^{-5}\allowbreak x^{6}-7.\,\allowbreak 781\,7\times
10^{-4}x^{5}-7.\,\allowbreak 206\,3\times 10^{-4}x^{4}-\allowbreak
3.\,\allowbreak 549\times 10^{-3}x^{3}+5.\,\allowbreak 790\,0\times
10^{-3}x^{2}+0.049\,6\allowbreak x-1.\,\allowbreak 798\,4\times 10^{-3}=0$,
Solution is: $5.\,\allowbreak 450\,6,4.\,\allowbreak 735\,4+\allowbreak
2.\,\allowbreak 080\,0i,4.\,\allowbreak 735\,4-\allowbreak 2.\,\allowbreak
080\,0i,2.\,\allowbreak 793\,9,\allowbreak 3.\,\allowbreak 610\,9\times
10^{-2},-0.172\,68+\allowbreak 2.\,\allowbreak
723\,8i,-0.172\,68-\allowbreak 2.\,\allowbreak 723\,8i,-2.\,\allowbreak
744\,5,-\allowbreak 4.\,\allowbreak 604\,3+2.\,\allowbreak 316\,5\allowbreak
i,-4.\,\allowbreak 604\,3-2.\,\allowbreak 316\,5\allowbreak
i,-5.\,\allowbreak 806\,0,-21.\,\allowbreak 943,-\allowbreak
29.\,\allowbreak 487$}

\subsection{$l(5.\,\allowbreak 450\,6)=\allowbreak 2.\,\allowbreak 281\,9$}

\subsection{$l(2.\,\allowbreak 793\,9)=\allowbreak 3.\,\allowbreak 103\,9$}

\subsection{$l(3.\,\allowbreak 610\,9\times 10^{-2})=\allowbreak
3.\,\allowbreak 209\,1$}

\subsection{$l(-2.\,\allowbreak 744\,5)=\allowbreak 3.\,\allowbreak 210\,6$}

\subsection{$l(-5.\,\allowbreak 806\,0)=\allowbreak 2.\,\allowbreak 807\,6$}

\subsection{$n=20$}

\subsection{$S(n)=\frac{\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))^{2}-f(g(4j+1))^{2}+2f(g(4j+2))^{2})=%
\allowbreak 10.\,\allowbreak 338\protect\pi $ = $32.\,\allowbreak 478$}

\subsection{$E(n)=\frac{\protect\pi }{4}\protect\int%
_{a}^{b}f(x)^{2}dx-S(n)=9.\,\allowbreak 961\protect\pi -10.\,\allowbreak 338%
\protect\pi =\allowbreak -0.377\,\protect\pi =-1.\,\allowbreak 184\,4$}

\subsection{$\frac{\protect\pi }{4}\frac{7h^{4}}{90}(b-a)\times
3.\,\allowbreak 210\,6=\allowbreak 0.153\,86\protect\pi =\allowbreak
0.483\,37$}

\bigskip

\subsection{$n=40$}

\subsection{$S(n)=\frac{\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))^{2}-f(g(4j+1))^{2}+2f(g(4j+2))^{2})=%
\allowbreak 10.\,\allowbreak 233\protect\pi =\allowbreak 32.\,\allowbreak
148 $}

\subsection{$E(n)=\frac{\protect\pi }{4}\protect\int_{a}^{b}f(x)^{2}dx-S(n)=%
\allowbreak 9.\,\allowbreak 961\protect\pi -10.\,\allowbreak 233\protect\pi %
=\allowbreak -0.272\,\protect\pi =\allowbreak -0.854\,51$}

\subsection{\protect\bigskip $\frac{\protect\pi }{4}\frac{7h^{4}}{90}%
(b-a)\times 3.\,\allowbreak 210\,6=\allowbreak 0.153\,86\protect\pi %
=\allowbreak 1.\,\allowbreak 158\,3\times 10^{-2}\protect\pi =\allowbreak
3.\,\allowbreak 638\,9\times 10^{-2}$}

\bigskip

\subsection{$n=80$}

\subsection{$S(n)=\frac{\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))^{2}-f(g(4j+1))^{2}+2f(g(4j+2))^{2})=%
\allowbreak 10.\,\allowbreak 116\protect\pi $ = $31.\,\allowbreak 78$}

\subsection{$E(n)=S(n)-\frac{\protect\pi }{4}\protect\int_{a}^{b}f(x)^{2}dx=%
\allowbreak \allowbreak 10.\,\allowbreak 116\protect\pi -\allowbreak
9.\,\allowbreak 961\protect\pi =\allowbreak 0.155\,\protect\pi =\allowbreak
0.486\,95$}

\bigskip

\subsection{$n=160$}

\subsection{$S(n)=\frac{\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2f(g(4j))^{2}-f(g(4j+1))^{2}+2f(g(4j+2))^{2})=%
\allowbreak 10.\,\allowbreak 043\protect\pi =\allowbreak 31.\,\allowbreak
551 $}

\subsection{$E(n)=S(n)-\frac{\protect\pi }{4}\protect\int_{a}^{b}f(x)^{2}dx=%
\allowbreak \allowbreak \allowbreak 10.\,\allowbreak 043\protect\pi %
-\allowbreak 9.\,\allowbreak 961\protect\pi =\allowbreak 0.082\,\protect\pi %
=\allowbreak 0.257\,61$}

\bigskip $x=\ln (%
\begin{array}{c}
20 \\ 
40 \\ 
80 \\ 
160%
\end{array}%
)=\allowbreak 
\begin{array}{c}
2.\,\allowbreak 995\,7 \\ 
3.\,\allowbreak 688\,9 \\ 
4.\,\allowbreak 382 \\ 
5.\,\allowbreak 075\,2%
\end{array}%
\allowbreak $

$y=\ln (%
\begin{array}{c}
32.\,\allowbreak 478 \\ 
\allowbreak 32.\,\allowbreak 148 \\ 
31.\,\allowbreak 78 \\ 
\allowbreak 31.\,\allowbreak 551%
\end{array}%
)=\allowbreak 
\begin{array}{c}
3.\,\allowbreak 480\,6 \\ 
3.\,\allowbreak 470\,4 \\ 
3.\,\allowbreak 458\,8 \\ 
3.\,\allowbreak 451\,6%
\end{array}%
\allowbreak $

\bigskip $%
\begin{array}{cc}
x & y \\ 
2.\,\allowbreak 995\,7 & 3.\,\allowbreak 480\,6 \\ 
3.\,\allowbreak 688\,9 & 3.\,\allowbreak 470\,4 \\ 
4.\,\allowbreak 094\,3 & 3.\,\allowbreak 458\,8 \\ 
4.\,\allowbreak 382 & 3.\,\allowbreak 451\,6%
\end{array}%
$, Polynomial fit: $y=3.\,\allowbreak 544\,9-2.\,\allowbreak 099\,0\times
10^{-2}\allowbreak x$\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special%
{language "Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth
0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine
"MuPAD";xmin "2";xmax "5";xviewmin "2";xviewmax "5";yviewmin "2";yviewmax
"4";viewset"XY";rangeset"X";plottype 4;axesFont "Times New
Roman,12,0000000000,useDefault,normal";numpoints 100;plotstyle
"patchnogrid";axesstyle "normal";axestips FALSE;xis \TEXUX{x};var1name
\TEXUX{$x$};function
\TEXUX{$\MATRIX{2,4}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,,,,}\CELL{2.\,%
\allowbreak 995\,7}\CELL{3.\,\allowbreak 480\,6}\CELL{3.\,\allowbreak
688\,9}\CELL{3.\,\allowbreak 470\,4}\CELL{4.\,\allowbreak
094\,3}\CELL{3.\,\allowbreak 458\,8}\CELL{4.\,\allowbreak
382}\CELL{3.\,\allowbreak 451\,6}$};linecolor "black";linestyle 1;pointplot
TRUE;pointstyle "circle";linethickness 1;lineAttributes "Solid";curveColor
"[flat::RGB:0000000000]";curveStyle "Point";function \TEXUX{$3.\,\allowbreak
544\,9-2.\,\allowbreak 099\,0\times 10^{-2}\allowbreak x$};linecolor
"black";linestyle 1;pointstyle "point";linethickness 1;lineAttributes
"Solid";var1range "2,5";num-x-gridlines 100;curveColor
"[flat::RGB:0000000000]";curveStyle "Line";VCamFile
'RMKL0U02.xvz';valid_file "T";tempfilename
'RMKL0U02.wmf';tempfile-properties "XPR";}}

\subsection{$\ln (|E|)=3.\,\allowbreak 544\,9-2.\,\allowbreak 099\,0\times
10^{-2}\ln (n)$}

\subsection{$\exp (3.\,\allowbreak 544\,9-2.\,\allowbreak 099\,0\times
10^{-2}\ln (n))=\allowbreak \frac{34.\,\allowbreak 636}{n^{0.020\,99}}$}

==============================================================3=============================================================

\bigskip

\subsection{$\protect\int_{a}^{b}2\protect\pi r(x)ds=\protect\int_{a}^{b}%
\protect\pi f(x)\frac{1}{2}\protect\sqrt{4+f^{\prime }(x)^{2}}dx=\frac{%
\protect\pi }{2}\protect\int_{a}^{b}f(x)\protect\sqrt{4+f^{\prime }(x)^{2}}%
dx=\allowbreak 74.\,\allowbreak 293$}

\subsection{$ds=\protect\sqrt{dx^{2}+dy^{2}}=\protect\sqrt{1+\left( \frac{dy%
}{dx}\right) ^{2}}dx=\protect\sqrt{1+r^{\prime }(x)^{2}}dx=\protect\sqrt{1+%
\frac{1}{4}f^{\prime }(x)^{2}}dx=\frac{1}{2}\protect\sqrt{4+f^{\prime
}(x)^{2}}dx$}

\subsection{$k(x)=f(x)\protect\sqrt{4+f^{\prime }(x)^{2}}$}

\bigskip

\bigskip

\subsection{$n=20$}

\subsection{$S(n)=\frac{2\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2k(g(4j))-k(g(4j+1))+2\ast k(g(4j+2)))=79.\,\allowbreak
106\allowbreak $}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}\protect\pi f(x)\frac{1}{2}%
\protect\sqrt{4+f^{\prime }(x)^{2}}dx=79.\,\allowbreak 106\allowbreak
-\allowbreak \allowbreak 74.\,\allowbreak 293=\allowbreak 4.\,\allowbreak
813 $}

\bigskip

\subsection{$n=40$}

\subsection{$S(n)=\frac{2\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2k(g(4j))-k(g(4j+1))+2\ast k(g(4j+2)))=\allowbreak
77.\,\allowbreak 073$}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}\protect\pi f(x)\frac{1}{2}%
\protect\sqrt{4+f^{\prime }(x)^{2}}dx=\allowbreak 77.\,\allowbreak
073\allowbreak -\allowbreak \allowbreak 74.\,\allowbreak 293=\allowbreak
2.\,\allowbreak 78$}

\bigskip

\subsection{$n=80$}

\subsection{$S(n)=\frac{2\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2k(g(4j))-k(g(4j+1))+2\ast k(g(4j+2)))=\allowbreak
75.\,\allowbreak 770$}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}\protect\pi f(x)\frac{1}{2}%
\protect\sqrt{4+f^{\prime }(x)^{2}}dx=\allowbreak 75.\,\allowbreak
770-\allowbreak \allowbreak 74.\,\allowbreak 293=\allowbreak 1.\,\allowbreak
477$}

\bigskip

\subsection{$n=160$}

\subsection{$S(n)=\frac{2\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2k(g(4j))-k(g(4j+1))+2\ast k(g(4j+2)))=75.\,\allowbreak
053\allowbreak $}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}\protect\pi f(x)\frac{1}{2}%
\protect\sqrt{4+f^{\prime }(x)^{2}}dx=\allowbreak 75.\,\allowbreak
053\allowbreak -\allowbreak \allowbreak 74.\,\allowbreak 293=\allowbreak
0.76 $}

\bigskip

\subsection{$n=320$}

\subsection{$S(n)=\frac{2\protect\pi }{3}h\sum%
\limits_{j=0}^{n/4}(2k(g(4j))-k(g(4j+1))+2\ast k(g(4j+2)))=\allowbreak
74.\,\allowbreak 679$}

\subsection{$E(n)=S(n)-\protect\int_{a}^{b}\protect\pi f(x)\frac{1}{2}%
\protect\sqrt{4+f^{\prime }(x)^{2}}dx=\allowbreak 74.\,\allowbreak
679\allowbreak -\allowbreak \allowbreak 74.\,\allowbreak 293=\allowbreak
0.386\,$}

\subsection{$x=\ln (%
\protect\begin{array}{c}
20\protect \\ 
40\protect \\ 
80\protect \\ 
160\protect \\ 
320%
\protect\end{array}%
)=\allowbreak 
\protect\begin{array}{c}
2.\,\allowbreak 995\,7\protect \\ 
3.\,\allowbreak 688\,9\protect \\ 
4.\,\allowbreak 382\protect \\ 
5.\,\allowbreak 075\,2\protect \\ 
5.\,\allowbreak 768\,3%
\protect\end{array}%
\allowbreak $}

\subsection{$y=\ln (%
\protect\begin{array}{c}
79.\,\allowbreak 106\allowbreak \protect \\ 
77.\,\allowbreak 073\protect \\ 
\allowbreak 75.\,\allowbreak 770\protect \\ 
75.\,\allowbreak 053\allowbreak \protect \\ 
\allowbreak 74.\,\allowbreak 679%
\protect\end{array}%
)=\allowbreak 
\protect\begin{array}{c}
4.\,\allowbreak 370\,8\protect \\ 
4.\,\allowbreak 344\,8\protect \\ 
4.\,\allowbreak 327\,7\protect \\ 
4.\,\allowbreak 318\,2\protect \\ 
4.\,\allowbreak 313\,2%
\protect\end{array}%
\allowbreak $}

\subsection{\protect\bigskip $%
\protect\begin{array}{cc}
x & y\protect \\ 
2.\,\allowbreak 995\,7 & 4.\,\allowbreak 370\,8\protect \\ 
3.\,\allowbreak 688\,9 & 4.\,\allowbreak 344\,8\protect \\ 
4.\,\allowbreak 382 & 4.\,\allowbreak 327\,7\protect \\ 
5.\,\allowbreak 075\,2 & 4.\,\allowbreak 318\,2\protect \\ 
5.\,\allowbreak 768\,3 & 4.\,\allowbreak 313\,2%
\protect\end{array}%
$\protect\FRAME{dtbpFX}{4.4996in}{3in}{0pt}{}{}{Plot}{\special{language
"Scientific Word";type "MAPLEPLOT";width 4.4996in;height 3in;depth
0pt;display "USEDEF";plot_snapshots TRUE;mustRecompute FALSE;lastEngine
"MuPAD";xmin "2";xmax "6";xviewmin "2";xviewmax "6";yviewmin "4";yviewmax
"5";viewset"XY";rangeset"X";plottype 4;axesFont "Times New
Roman,12,0000000000,useDefault,normal";numpoints 100;plotstyle
"patchnogrid";axesstyle "normal";axestips FALSE;xis \TEXUX{x};var1name
\TEXUX{$x$};function
\TEXUX{$\MATRIX{2,5}{c}\VR{,,c,,,}{,,c,,,}{,,,,,}\HR{,,,,,}\CELL{2.\,%
\allowbreak 995\,7}\CELL{4.\,\allowbreak 370\,8}\CELL{3.\,\allowbreak
688\,9}\CELL{4.\,\allowbreak 344\,8}\CELL{4.\,\allowbreak
382}\CELL{4.\,\allowbreak 327\,7}\CELL{5.\,\allowbreak
075\,2}\CELL{4.\,\allowbreak 318\,2}\CELL{5.\,\allowbreak
768\,3}\CELL{4.\,\allowbreak 313\,2}$};linecolor "black";linestyle
1;pointplot TRUE;pointstyle "circle";linethickness 1;lineAttributes
"Solid";curveColor "[flat::RGB:0000000000]";curveStyle "Point";function
\TEXUX{$4.\,\allowbreak 424\,6-2.\,\allowbreak 045\,7\times
10^{-2}\allowbreak x$};linecolor "black";linestyle 1;pointstyle
"point";linethickness 1;lineAttributes "Solid";var1range
"2,6";num-x-gridlines 100;curveColor "[flat::RGB:0000000000]";curveStyle
"Line";VCamFile 'RMKL0V03.xvz';valid_file "T";tempfilename
'RMKL0V03.wmf';tempfile-properties "XPR";}}, Polynomial fit: $%
y=4.\,\allowbreak 424\,6-2.\,\allowbreak 045\,7\times 10^{-2}\allowbreak x$}

\bigskip 

\subsection{$\ln (|E|)=4.\,\allowbreak 424\,6-2.\,\allowbreak 045\,7\times
10^{-2}\ln (n)$}

\subsection{$\exp (4.\,\allowbreak 424\,6-2.\,\allowbreak 045\,7\times
10^{-2}\ln (n))=\allowbreak \frac{83.\,\allowbreak 479}{n^{2.\,\allowbreak
045\,7\times 10^{-2}}}$}

\end{document}