%% %% An UIT Edition example %% %% Example 04-20-7 on page 71. %% %% Copyright (C) 2010 Herbert Voss %% %% It may be distributed and/or modified under the conditions %% of the LaTeX Project Public License, either version 1.3 %% of this license or (at your option) any later version. %% %% See http://www.latex-project.org/lppl.txt for details. %% %% %% ==== % Show page(s) 1 %% \documentclass[]{exaarticle} \pagestyle{empty} \setlength\textwidth{375.57637pt} \AtBeginDocument{\setlength\parindent{0pt} \usepackage[utf8]{inputenc}} \usepackage[english]{babel} \StartShownPreambleCommands \usepackage{array,graphicx} \newcommand*\diff{\mathop{}\!\mathrm{d}} \StopShownPreambleCommands \begin{document} \def\arraystretch{2.1} \rotatebox{90}{% $\begin{array}{>{\displaystyle}r@{\kern1.4pt}>{\displaystyle}l} A=\lim _{n\rightarrow\infty}U &= \lim_{n\rightarrow\infty}\sum^{n-1}_{i=0}\left(\Delta x\times f(a+i\times\Delta x\right) \\ &= \lim_{n\rightarrow\infty}\left(\Delta x\times f(a)+\Delta x\times f(a+\Delta x)+\Delta x\times f(a+2\times\Delta x)+\Delta x\times f(a+3\times \Delta x)+\ldots +\Delta x\times f(a+(n-1)\times \Delta x)\right) \\ &= \lim_{n\rightarrow\infty}\Delta x\times\left(f(a)+f(a+\Delta x)+f(a+2\times \Delta x)+f(a+3\times\Delta x)+\ldots+f(a+(n-1)\times\Delta x)\right) \\ &= \lim_{n\rightarrow\infty}\Delta x\left(a^2+\left(a+\Delta x\right)^2+\left(a+2\times\Delta x\right)^2+\left( a+3\times \Delta x\right)^2+\ldots +\left(a+(n-1)\times \Delta x\right)^2\right) \\ &= \lim_{n\rightarrow\infty}\Delta x\left(a^2+\left(a^2+2a\Delta x+\left(\Delta x\right)^2\right)+\left(a^2+2\times2a\Delta x+2^2\left(\Delta x\right)^2\right)+\left(a^2+2\times3a\Delta x+3^2\left( \Delta x\right)^2\right)+\ldots\right.\\ & \qquad\left.\quad+\left(a^2+2\times(n-1)a\Delta x+(n-1)^2\left(\Delta x\right)^2\right)\right) \\ &= \lim_{n\rightarrow\infty}\Delta x\left(na^2+2a\Delta x\left( 1+2+3+\ldots+(n-1)\right)+\left(\Delta x\right)^2\left(1^2+2^2+3^2+\ldots+(n-1)^2\right)\right) \\ &= \lim_{n\rightarrow\infty}\Delta x\left(na^2+2a\Delta x\frac{n(n-1)}2+\left(\Delta x\right)^2\frac{n(2n-1)(n-1)}{6}\right) \\ &= \lim_{n\rightarrow\infty}\frac{b-a}{n}\left(na^2+2a\frac{b-a}{n}\frac{n(n-1)}2+\left(\frac{b-a}{n}\right)^2\frac{n(2n-1)(n-1)}{6}\right) \\ &= \lim_{n\rightarrow\infty}\frac{b-a}{n}\left(na^2+a(b-a)(n-1)+\frac{(b-a)^2}{n}\frac{(2n-1)(n-1)}{6}\right) \\ &= \lim_{n\rightarrow\infty}(b-a)\left(a^2+a(b-a)\frac{n-1}{n}+\frac{(b-a)^2}{n^2}\frac{(2n-1)(n-1)}{6}\right) \\ &= \lim_{n\rightarrow\infty}(b-a)\left(a^2+a(b-a)\left(1-\underbrace{\frac{1}{n}}\right)+(b-a)^2\frac{1}{6}\left(2-\underbrace{\frac{3}{n}}+\underbrace{\frac{1}{n^2}}\right)\right) \\ \multicolumn{2}{r@{}}{\mbox{null series for }n\rightarrow\infty} \end{array}$} \end{document}