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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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