Warning: this is an htmlized version!
The original is here, and
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% (find-angg "LATEX/2019-1-C2-material.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2019-1-C2-material.tex" :end))
% (defun d () (interactive) (find-pdf-page         "~/LATEX/2019-1-C2-material.pdf"))
% (defun d () (interactive) (find-pdftools-page    "~/LATEX/2019-1-C2-material.pdf"))
% (defun e () (interactive) (find-LATEX "2019-1-C2-material.tex"))
% (defun u () (interactive) (find-latex-upload-links "2019-1-C2-material"))
% (find-xpdfpage "~/LATEX/2019-1-C2-material.pdf")
% (find-sh0 "cp -v  ~/LATEX/2019-1-C2-material.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2019-1-C2-material.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2019-1-C2-material.pdf
%               file:///tmp/2019-1-C2-material.pdf
%           file:///tmp/pen/2019-1-C2-material.pdf
% http://angg.twu.net/LATEX/2019-1-C2-material.pdf

% «.defs»		(to "defs")
% «.defs-int-subst»	(to "defs-int-subst")
% «.int-subst»		(to "int-subst")
% «.trab-area-superfs»	(to "trab-area-superfs")

\documentclass[oneside]{book}
\usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref")
%\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{pict2e}
\usepackage{xcolor}                % (find-es "tex" "xcolor")
%\usepackage{color}                % (find-LATEX "edrx15.sty" "colors")
%\usepackage{colorweb}             % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof}   % For derivation trees ("%:" lines)
%\input diagxy        % For 2D diagrams ("%D" lines)
%\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%\def\expr#1{\directlua{output(tostring(#1))}}
%\def\eval#1{\directlua{#1}}
%
\usepackage{edrx15}               % (find-LATEX "edrx15.sty")
\input edrxaccents.tex            % (find-LATEX "edrxaccents.tex")
\input edrxchars.tex              % (find-LATEX "edrxchars.tex")
\input edrxheadfoot.tex           % (find-LATEX "edrxheadfoot.tex")
\input edrxgac2.tex               % (find-LATEX "edrxgac2.tex")
%
% (find-angg ".emacs.papers" "latexgeom")
% (find-LATEXfile "2016-2-GA-VR.tex" "{geometry}")
% (find-latexgeomtext "total={6.5in,8.75in},")
\usepackage[%paperwidth=11.5cm, paperheight=9cm,
            %total={6.5in,4in},
            %textwidth=4in,  paperwidth=4.5in,
            %textheight=5in, paperheight=4.5in,
            %a4paper,
            top=2.5cm, bottom=2.5cm, left=2.5cm, right=2.5cm, includefoot
           ]{geometry}
\begin{document}

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

\directlua{dofile "edrxtikz.lua"} % (find-LATEX "edrxtikz.lua")
\directlua{dofile "edrxpict.lua"} % (find-LATEX "edrxpict.lua")
%L V.__tostring = function (v) return format("(%.3f,%.3f)", v[1], v[2]) end



% (find-LATEX "2017-2-C2-material.tex" "integracao-por-substituicao")



%  ____        __     
% |  _ \  ___ / _|___ 
% | | | |/ _ \ |_/ __|
% | |_| |  __/  _\__ \
% |____/ \___|_| |___/
%                     
% «defs» (to ".defs")

\def\subst#1{\left[\sm{#1}\right]}
\def\Subst#1{\left[\mat{#1}\right]}
\def\ddx{\frac{d}{dx}}

\def\pfo#1{\ensuremath{(\mathsf{#1})}}
\def\D#1{\displaystyle #1}

% Difference with mathstrut
\def\Difms #1#2#3{\left. \mathstrut #3 \right|_{s=#1}^{s=#2}}
\def\Difmu #1#2#3{\left. \mathstrut #3 \right|_{u=#1}^{u=#2}}
\def\Difmx #1#2#3{\left. \mathstrut #3 \right|_{x=#1}^{x=#2}}
\def\Difmth#1#2#3{\left. \mathstrut #3 \right|_{θ=#1}^{θ=#2}}

\def\iequationbox#1#2{
    \left(
    \begin{array}{rcl}
    \D{ #1 } &=& \D{ #2 } \\
    \end{array}
    \right)
  }
\def\isubstbox#1#2#3#4#5{{
    \def\veq{\rotatebox{90}{$=$}}
    \def\ph{\phantom}
    \left(
    \begin{array}{rcl}
    \D{ #1 } &=& \D{ #2 } \\
    {\veq#3} \\
    \D{ #4 } &=& \D{ #5 } \\
    \end{array}
    \right)
  }}


% «defs-int-subst» (to ".defs-int-subst")
% Definição das fórmulas para integração por substituição.
% Algumas são pmatrizes 3x3 usando isubstbox.

\def\TFCtwo{
  \iequationbox {\Intx{a}{b}{F'(x)}}
                {\Difmx{a}{b}{F(x)}}
}
\def\TFCtwoI{
  \iequationbox {\intx{F'(x)}}
                {F(x)}
}

\def\Sone{
  \isubstbox
    {\Difmx{a}{b}{f(g(x))}}  {\Intx{a}{b}{f'(g(x))g'(x)}}
    {\ph{mmm}}
    {\Difmu{g(a)}{g(b)}{f(u)}} {\Intu{g(a)}{g(b)}{f'(u)}}
}
\def\SoneI{
  \isubstbox
    {f(g(x))} {\intx{f'(g(x))g'(x)}}
    {\ph{m}}
    {f(u)}    {\intx{f'(u)}}
}

\def\Stwo{
  \isubstbox
    {\Difmx{a}{b}{F(g(x))}}   {\Intx{a}{b}{f(g(x))g'(x)}}
    {\ph{mmm}}
    {\Difmu{g(a)}{g(b)}{F(u)}}  {\Intu{g(a)}{g(b)}{f(u)}}
}
\def\StwoI{
  \isubstbox
    {F(g(x))}  {\intx{f(g(x))g'(x)}}
    {\ph{m}}
    {F(u)}     {\intu{f(u)}}
}

\def\Sthree{
  \iequationbox {\Intx{a}{b}{f(g(x))g'(x)}}
                {\Intu{g(a)}{g(b)}{f(u)}}
}
\def\SthreeI{
  \iequationbox {\intx{f(g(x))g'(x)}}
                {\int{f(u)}}
  (u=g(x))
}





%  ___       _               _         _   
% |_ _|_ __ | |_   ___ _   _| |__  ___| |_ 
%  | || '_ \| __| / __| | | | '_ \/ __| __|
%  | || | | | |_  \__ \ |_| | |_) \__ \ |_ 
% |___|_| |_|\__| |___/\__,_|_.__/|___/\__|
%                                          
% «int-subst» (to ".int-subst")
% (c2m191p 1 "int-subst")
% (c2m191    "int-subst")

{\bf Integração por substituição}

\pfo{S1}, \pfo{S2}, \pfo{S3}: substituição na integral definida (mais
concreta),

\pfo{S1I}, \pfo{S2I}, \pfo{S3I}: substituição na integral indefinida
(mais abstrata).

Os livros costumam começar pela fórmula $\pfo{SI3}$, que é a mais
abstrata de todas...

Nós vamos seguir um caminho bem diferente, e vamos tratar as fórmulas

\pfo{TFC2I}, \pfo{S1I}, \pfo{S2I}, \pfo{S3I} como {\sl abreviações} para as fórmulas

\pfo{TFC2}, \pfo{S1}, \pfo{S2}, \pfo{S3}.



$$\begin{array}[t]{rcl}
  \text{Fórmulas}:        \\[5pt]
  \pfo{TFC2} &=& \TFCtwo  \\ \\
    \pfo{S1} &=& \Sone    \\ \\
    \pfo{S2} &=& \Stwo    \\ \\
    \pfo{S3} &=& \Sthree  \\ \\
 \pfo{TFC2I} &=& \TFCtwoI \\ \\
   \pfo{S1I} &=& \SoneI   \\ \\
   \pfo{S2I} &=& \StwoI   \\ \\
   \pfo{S3I} &=& \SthreeI
  \end{array}
  %
  \quad
  %
  \begin{tabular}[t]{l}
    Exercícios: \\[5pt]
    a) $\pfo{TFC2} \subst{F(x):=-\cos x \\ a:=0 \\ b:=π}$      \\
    b) $\pfo{TFC2} \subst{F(x):=\cos x}$                       \\
    c) $\pfo{TFC2} \subst{F(x):=\cos x} \subst{a:=0 \\ b:=π}$  \\
    d) $\pfo{TFC2} \subst{F(x):=\cos x} \subst{a:=π \\ b:=2π}$ \\
    e) $\pfo{TFC2} \subst{F(x):=\frac12 x^2 \\ a:=0 \\ b:=4 }$ \\
    f) $\pfo{TFC2} \subst{F(x):=\frac13 x^3 \\ a:=0 \\ b:=2 }$ \\
    g) $f(g(x)) \subst{f(u):=\sen u \\ g(x) := 4x} $\\
    h) $(f'(g(x))g'(x)) \subst{f(u):=\sen u \\ g(x) := 4x} $\\
    %
    % (find-angg ".emacs" "c2q182")
    % (c2q182  6 "20180822" "TFC2; substituição")
    i) $\pfo{S1} \subst{
                    f(u) := \sen u \\
                    g(x) := 3x+4 \\
                    a := 1 \\
                    b := 2 \\
                 }$
    \\
    j) $\pfo{S2} \subst{
                    F(u) := \sen u \\
                    f(u) := \cos u \\
                    g(x) := 3x+4 \\
                    a := 1 \\
                    b := 2 \\
                 }$
    \\
    k) $\pfo{S2} \subst{
                    f(u) := \cos u \\
                    g(x) := 3x+4 \\
                    a := 1 \\
                    b := 2 \\
                 }$
    \\
    l) $\pfo{S2} \subst{
                    f(u) := \sqrt{u} \\
                    g(x) := 3x+4 \\
                    a := 1 \\
                    b := 2 \\
                 }$
    \\
    m) $\pfo{S3} \subst{
                    f(u) := \sqrt{u} \\
                    g(x) := 3x+4 \\
                    a := 1 \\
                    b := 2 \\
                 }$
    \\
    \\
    i') $\pfo{S1I} \subst{
                    f(u) := \sen u \\
                    g(x) := 3x+4 \\
                    a := 1 \\
                    b := 2 \\
                 }$
    \\
    i'') $\pfo{S1I} \subst{
                    f(u) := \sen u \\
                    g(x) := 3x+4 \\
                    % a := 1 \\
                    % b := 2 \\
                 }$
    \\
    k') $\pfo{S2I} \subst{
                    f(u) := \cos u \\
                    g(x) := 3x+4 \\
                    a := 1 \\
                    b := 2 \\
                 }$
    \\
    k'') $\pfo{S2I} \subst{
                    f(u) := \cos u \\
                    g(x) := 3x+4 \\
                    % a := 1 \\
                    % b := 2 \\
                 }$
    \\
    m') $\pfo{S3I} \subst{
                    f(u) := \sqrt{u} \\
                    g(x) := 3x+4 \\
                 }$
    \\
    % (find-angg ".emacs" "c2q182")
  \end{tabular}
$$



\newpage

%     _                                                __     
%    / \   _ __ ___  __ _   ___ _   _ _ __   ___ _ __ / _|___ 
%   / _ \ | '__/ _ \/ _` | / __| | | | '_ \ / _ \ '__| |_/ __|
%  / ___ \| | |  __/ (_| | \__ \ |_| | |_) |  __/ |  |  _\__ \
% /_/   \_\_|  \___|\__,_| |___/\__,_| .__/ \___|_|  |_| |___/
%                                    |_|                      
%
% «trab-area-superfs»  (to ".trab-area-superfs")
% (c2m191p 99 "trab-area-superfs")
% (c2m191a    "trab-area-superfs")

\def\AreaEntre{\textsf{ÁreaEntre}}


Trabalho sobre áreas de superfícies de revolução

Vale 0.5 pontos na VR ou na VS (que vão ter questões sobre isso),

o que for mais vantajoso pra vocês.

\msk

Sejam:

$P(x,y) = (x,y)$,

$C(x,R) = \setofst{(x,y,z)∈\R^3}{x^2 + y^2 = R^2}$.

\msk

1) Calcule as distâncias:

a) $d(P(4,2),P(7,2))$

b) $d(P(4,3),P(7,3))$

c) $d(P(4,2),P(4,3))$

d) $d(P(4,3),P(7,2))$

\msk

2) Calcule as áreas dos pedaços de cones entre:

a) $C(4,2)$ e $C(7,2)$

b) $C(4,3)$ e $C(7,3)$

c) $C(4,2)$ e $C(4,3)$

\msk

3) Represente graficamente os segmentos 1a, 1b, 1c, 1d.

\msk

4) Encontre no olhômetro (1d)/(1a), (1d)/(1b), (1d)/(1c).

(Em sala nós chamamos eles de ``fatores multiplicadores'').

\msk

5) Será que os ``fatores multiplicadores'' que você encontrou na 4
servem para calcular a área do pedaço de cone entre $C(4,3)$ e
$C(7,2)$? Não examente, mas vamos fingir que sim... qual {\sl seria} o
fator multiplicador

a) de $\AreaEntre(C(4,2),C(7,2))$ para $\AreaEntre(C(4,3),C(7,2))$?

b) de $\AreaEntre(C(4,3),C(7,3))$ para $\AreaEntre(C(4,3),C(7,2))$?

c) de $\AreaEntre(C(4,2),C(4,3))$ para $\AreaEntre(C(4,3),C(7,2))$?

\msk

6) Usando os fatores multiplicadores do item anterior calcule:

a) $\AreaEntre(C(4,2),C(7,2))$ (item 2a!) e a partir dela $\AreaEntre(C(4,3),C(7,2))$

b) $\AreaEntre(C(4,3),C(7,3))$ (item 2b!) e a partir dela $\AreaEntre(C(4,3),C(7,2))$

c) $\AreaEntre(C(4,2),C(4,3))$ (item 2c!) e a partir dela $\AreaEntre(C(4,3),C(7,2))$

\msk

7) Use uma calculadora pra calcular numericamente os resultados dos
itens 6a, 6b, 6c.

\msk

8) Agora vamos generalizar o problema 5. Qual é o ``fator multiplicador''

a) de $\AreaEntre(C(x_0,y_0),C(x_1,y_0))$ para $\AreaEntre(C(x_0,y_0),C(x_1,y_1))$?

b) de $\AreaEntre(C(x_0,y_1),C(x_1,y_1))$ para $\AreaEntre(C(x_0,y_0),C(x_1,y_1))$?

c) de $\AreaEntre(C(x_0,y_0),C(x_0,y_1))$ para $\AreaEntre(C(x_0,y_0),C(x_1,y_1))$?

\msk

9) Use os fatores multiplicadores do item anterior para calcular:

a) $\AreaEntre(C(x_0,y_0),C(x_1,y_0))$ (item 8a!) e a partir dela $\AreaEntre(C(x_0,y_0),C(x_1,y_1))$

b) $\AreaEntre(C(x_0,y_1),C(x_1,y_1))$ (item 8b!) e a partir dela $\AreaEntre(C(x_0,y_0),C(x_1,y_1))$

c) $\AreaEntre(C(x_0,y_0),C(x_0,y_1))$ (item 8c!) e a partir dela $\AreaEntre(C(x_0,y_0),C(x_1,y_1))$

\msk

10) Simplifique as respostas dos itens 9a, 9b e 9c usando:
$Δx=x_1-x_0$, $Δy=y_1-y_0$, $y_x = \frac{Δy}{Δy}$.










\bsk









\end{document}




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% coding: utf-8-unix
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