%% This document created by Scientific Notebook (R) Version 3.5 %% Starting shell: article \documentclass{article} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amssymb} %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Version=5.00.0.2570} %TCIDATA{} %TCIDATA{Created=Wednesday, February 10, 1999 13:29:48} %TCIDATA{LastRevised=Sunday, February 13, 2005 17:06:15} %TCIDATA{} %TCIDATA{} %TCIDATA{CSTFile=On line bluem.cst} %TCIDATA{PageSetup=72,72,72,72,0} %TCIDATA{Counters=arabic,1} %TCIDATA{AllPages= %H=36 %F=36,\PARA{038
\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Integr\U{e1}lny po\U{10d}et - V\U{fd}po\U{10d}et integr\U{e1}lov\dotfill \thepage }} %} \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]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \author{A. U. Thor} \title{Lab Report} \date{The Date } \maketitle \begin{abstract} A Laboratory report created with Scientific Notebook \end{abstract} \section{Integr\'{a}lny po\v{c}et} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{\hyperref{Obsah kapitoly}{}{}{Ma5.tex}}{}{}{Ma5.tex}}% %BeginExpansion \msihyperref{% \msihyperref{Obsah kapitoly}{}{}{Ma5.tex}}{}{}{Ma5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma54.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma54.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma56.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma56.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C5.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \subsection{V\'{y}po\v{c}et integr\'{a}lov.} Budeme sa koncentrova\v{t} na dvojn\'{e} integr\'{a}ly, preto\v{z}e tento pr% \'{\i}pad n\'{a}m vysvetl\'{\i} v\v{s}eobecn\'{u} met\'{o}du. \begin{definition} a) Nech $\alpha ,\beta $ s\'{u} spojit\'{e} re\'{a}lne funkcie $\alpha ,\beta :\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$ a nech $% \forall x\mathbf{\in }\left\langle a,b\right\rangle $ je $\alpha (x)\leq \beta (x).$Mno\v{z}inu $A=\{(x,y)\in \mathbf{R}^{2};a\leq x\leq b,\alpha (x)\leq y\leq \beta (x)\}$ naz\'{y}vame \label{3}\emph{element\'{a}rna oblas% \v{t} typu} $[x,y].$ b) Nech $\gamma ,\delta $ s\'{u} spojit\'{e} re\'{a}lne funkcie $\gamma ,\delta :\left\langle c,d\right\rangle \longrightarrow \mathbf{R}$ a nech $% \forall y\mathbf{\in }\left\langle c,d\right\rangle $ je $\gamma (y)\leq \delta (y).$ Mno\v{z}inu $A=\{(x,y)\in \mathbf{R}^{2};\,c\leq y\leq d,\gamma (y)\leq x\leq \delta (y)\}$ naz\'{y}vame \label{4}\emph{element\'{a}rna oblas% \v{t} typu} $[y,x].$ \end{definition} \begin{theorem} \label{2}Nech $A$ je element\'{a}rna oblas\v{t} typu $[x,y]$ ($[y,x]$). Predpokladajme, \v{z}e $f:A\longrightarrow \mathbf{R}$ je ohrani\v{c}en\'{a} a spojit\'{a} s v\'{y}nimkou (najviac) hranice $A$ a pre najviac kone\v{c}n% \'{y} po\v{c}et bodov z vn\'{u}tra $A.$ Potom \[ \int_{A}f=\iint_{A}f(x,y)dxdy=\int_{a}^{b}\left( \int_{\alpha \left( x\right) }^{\beta \left( x\right) }f(x,y)dy\right) dx \]% \[ \left[ \int_{A}f=\iint_{A}f(x,y)dxdy=\int_{c}^{d}\left( \int_{\gamma \left( y\right) }^{\delta \left( y\right) }f(x,y)dx\right) dy\right] . \] \end{theorem} \begin{tabular}{|c|} \hline {\small %TCIMACRO{\hyperref{D\^{o}kaz}{}{}{DO551.tex}}% %BeginExpansion \msihyperref{D\^{o}kaz}{}{}{DO551.tex}% %EndExpansion } \\ \hline \end{tabular}% \label{1} \begin{example} Nech $A$ je trojuholn\'{\i}k ohrani\v{c}en\'{y} \v{c}iarami $y=0,x=1,y=x.$ Vypo\v{c}\'{\i}tajte $\iint_{A}e^{x^{2}}dxdy.$ \end{example} \begin{solution} Najsk\^{o}r pop\'{\i}\v{s}eme element\'{a}rnu oblas\v{t} $A$ ako oblas\v{t} typu $\left[ y,x\right] .$ M\'{a}me: $A:0\leq y\leq 1,\,y\leq x\leq 1$% \[ \iint_{A}e^{x^{2}}dxdy=\int_{0}^{1}\left( \int_{y}^{1}e^{x^{2}}dx\right) dy. \]% Takto v\v{s}ak integr\'{a}l nevieme vypo\v{c}\'{\i}ta\v{t}. Ale ak budeme integrova\v{t} v opa\v{c}nom porad\'{\i} t.j. $A$ pop\'{\i}\v{s}eme ako element\'{a}rnu oblas\v{t} typu $\left[ x,y\right] :A:0\leq x\leq 1,0\leq y\leq x$ tak dostaneme:% \[ \iint_{A}e^{x^{2}}dxdy=\int_{0}^{1}\left( \int_{0}^{x}e^{x^{2}}dy\right) dx=\int_{0}^{1}\left( \left[ ye^{x^{2}}\right] _{0}^{x}\right) dx=\int_{0}^{1}xe_{{}}^{x^{2}}dx= \]% \[ =\left| \begin{tabular}{cc} $t=x^{2}$ & $dt=2xdx$ \\ $x=0\Longrightarrow t=0$ & $x=1\Longrightarrow t=1$% \end{tabular}% \right| =\frac{1}{2}\int_{0}^{1}e^{t}dt=\frac{e-1}{2}.\square \] \end{solution} Podobn\'{y}m sp\^{o}sobom ako pre pr\'{\i}pad dvoch premenn\'{y}ch by sme dok% \'{a}zali vetu pre v\'{y}po\v{c}et trojn\'{y}ch a viacn\'{a}sobn\'{y}ch integr\'{a}lov na element\'{a}rnych oblastiach. \begin{definition} Nech $A$ je element\'{a}rna oblas\v{t} typu $[x,y]$ ($[y,x]$), nech $g,h$ s% \'{u} spojit\'{e} funkcie $g,h:A\longrightarrow \mathbf{R}$ a nech $\forall (x,y)\in A$ je $g(x,y)\leq h(x,y).$ Mno\v{z}inu $B\subset \mathbf{R}^{3};\,$% \[ B=\left\{ (x,y,z)\mathbf{\in \mathbf{R}}^{3};\,(x,y)\in A,g(x,y)\leq z\leq h(x,y)\right\} \]% naz\'{y}vame \label{5}\emph{element\'{a}rna oblas\v{t} typu} $[x,y,z]$ ($% [y,x,z]$). Analogicky definujeme element\'{a}rne oblasti typu $% [x,z,y],\,[y,z,x],\dots $ \end{definition} \begin{theorem} \label{6}Nech $B\subset \mathbf{R}^{3}$ je element\'{a}rna oblas\v{t} typu $% [x,y,z].$ Predpokladajme, \v{z}e $f:B\longrightarrow \mathbf{R}$ je ohrani% \v{c}en\'{a} a spojit\'{a} s v\'{y}nimkou (najviac) hranice mno\v{z}iny $B$ a pre najviac kone\v{c}n\'{y} po\v{c}et bodov z vn\'{u}tra $B.$ Potom \[ \int_{B}f=\iiint_{B}f(x,y,z)dxdydz=\iint_{A}\left( \int_{g\left( x,y\right) }^{h\left( x,y\right) }f(x,y,z)dz\right) dxdy \] \end{theorem} \begin{example} Nech $A$ je \v{s}tvorsten $A=\left\{ \left( x,y,z\right) \in \mathbf{R}% ^{3};\,x\geq 0,y\geq 0,z\geq 0,z\leq 1-x-y\right\} .$ Vypo\v{c}\'{\i}tajte $% \iiint_{A}\left( 1-x\right) yzdxdydz.$ \end{example} \begin{solution} Najsk\^{o}r si zap\'{\i}\v{s}eme $A$ ako element\'{a}rnu oblas\v{t} typu $% \left[ x,y,z\right] $\ pomocou nerovnost\'{\i}: \[ A:0\leq x\leq 1,0\leq y\leq 1-x,0\leq z\leq 1-x-y, \]% potom m\'{a}me \[ \iiint_{A}\left( 1-x\right) yzdxdydz=\int_{0}^{1}\left( \int_{0}^{1-x}\left[ \int_{0}^{1-x-y}\left( 1-x\right) yzdz\right] dy\right) dx= \]% \[ =\int_{0}^{1}\left( \int_{0}^{1-x}\left( 1-x\right) y\left[ \frac{z^{2}}{2}% \right] _{0}^{1-x-y}dy\right) dx=\frac{1}{2}\int_{0}^{1}\left( \int_{0}^{1-x}\left( 1-x\right) y\left( 1-x-y\right) ^{2}dy\right) dx= \]% \[ =\frac{1}{2}\int_{0}^{1}\left( 1-x\right) \left( \int_{0}^{1-x}y\left( 1-x-y\right) ^{2}dy\right) dx= \]% \[ =\frac{1}{2}\int_{0}^{1}\left( 1-x\right) \left( \left[ \left( 1-x\right) ^{2}\frac{y^{2}}{2}-2\left( 1-x\right) \frac{y^{3}}{3}+\frac{y^{4}}{4}\right] _{0}^{1-x}\right) dx=\frac{1}{24}\int_{0}^{1}\left( 1-x\right) ^{5}dx=\frac{1% }{144}.\square \] \end{solution} \begin{example} Nech $A=\left\{ \left( x,y,z\right) \in \mathbf{R}^{3};\,\sqrt{x^{2}+y^{2}}% \leq z\leq \sqrt{2-x^{2}-y^{2}}\right\} .$ Vypo\v{c}\'{\i}tajte $% \iiint_{A}zdxdydz.$ \end{example} \begin{solution} Najsk\^{o}r zap\'{\i}\v{s}eme $A$ ako element\'{a}rnu oblas\v{t} pomocou nerovnost\'{\i}: \[ A:-1\leq x\leq 1,-\sqrt{1-x^{2}}\leq y\leq \sqrt{1-x^{2}},\sqrt{x^{2}+y^{2}}% \leq z\leq \sqrt{2-x^{2}-y^{2}}, \]% potom m\'{a}me \[ \iiint_{A}zdxdydz=\int_{-1}^{1}\left( \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}% \left[ \int_{\sqrt{x^{2}+y^{2}}}^{\sqrt{2-x^{2}-y^{2}}}z\right] dy\right) dx= \]% \[ =\int_{-1}^{1}\left( \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}\left[ \frac{% z^{2}}{2}\right] _{\sqrt{x^{2}+y^{2}}}^{\sqrt{2-x^{2}-y^{2}}}dy\right) dx=% \frac{1}{2}\int_{-1}^{1}\left( \int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}% }2\left( 1-x^{2}-y^{2}\right) dy\right) dx= \]% \[ =\int_{-1}^{1}\left[ \left( 1-x^{2}\right) y-\frac{y^{3}}{3}\right] _{-\sqrt{% 1-x^{2}}}^{\sqrt{1-x^{2}}}dx=\frac{4}{3}\int_{-1}^{1}\left( 1-x^{2}\right) ^{% \frac{3}{2}}dx= \] \[ =\left| \begin{tabular}{cc} $x=\sin u$ & $dx=\cos udu$ \\ $x=-1\Longrightarrow u=-\frac{\pi }{2}$ & $x=1\Longrightarrow u=\frac{\pi }{2% }$% \end{tabular}% \right| =\frac{4}{3}\int_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\cos ^{4}udu=% \frac{\pi }{2}.\square \] \end{solution} \begin{center} \begin{tabular}{|c|c|c|c|c|c|c|} \hline \textbf{% %TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}% %BeginExpansion \msihyperref{Obsah}{}{}{maiindex.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{\hyperref{Obsah kapitoly}{}{}{Ma5.tex}}{}{}{Ma5.tex}}% %BeginExpansion \msihyperref{% \msihyperref{Obsah kapitoly}{}{}{Ma5.tex}}{}{}{Ma5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma54.tex}}% %BeginExpansion \msihyperref{Predch\'{a}dzaj\'{u}ca str\'{a}nka}{}{}{Ma54.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma56.tex}}% %BeginExpansion \msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma56.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Ot\'{a}zky}{}{}{O5.tex}}% %BeginExpansion \msihyperref{Ot\'{a}zky}{}{}{O5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Cvi\v{c}enia}{}{}{C5.tex}}% %BeginExpansion \msihyperref{Cvi\v{c}enia}{}{}{C5.tex}% %EndExpansion } & \textbf{% %TCIMACRO{\hyperref{Index}{}{}{Glos.tex}}% %BeginExpansion \msihyperref{Index}{}{}{Glos.tex}% %EndExpansion } \\ \hline \end{tabular} \end{center} \rule{6.5in}{0.04in} \textsl{Matematick\'{a} anal\'{y}za II} \section{Integr\'{a}lny po\v{c}et} \end{document}