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\QTR{small}{Matematick\U{e1} anal\U{fd}za II online - Integr\U{e1}lny po\U{10d}et - Integrovanie cez n-rozmern\U{e9} kv\U{e1}dre\dotfill \thepage }}
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\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|}
\hline
\textbf{%
%TCIMACRO{\hyperref{Obsah}{}{}{maiindex.tex}}%
%BeginExpansion
\msihyperref{Obsah}{}{}{maiindex.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Obsah kapitoly}{}{}{Ma5.tex}}%
%BeginExpansion
\msihyperref{Obsah kapitoly}{}{}{Ma5.tex}%
%EndExpansion
} & \textbf{%
%TCIMACRO{\hyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma52.tex}}%
%BeginExpansion
\msihyperref{Nasleduj\'{u}ca str\'{a}nka}{}{}{Ma52.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}
\paragraph{Ciele}
Po pre\v{s}tudovan\'{\i} tejto \v{c}asti by ste mali by\v{t} schopn\'{\i}:
\begin{itemize}
\item vysvetli\v{t} pojem integrovate\v{l}nej funkcie viacer\'{y}ch premenn%
\'{y}ch,
\item op\'{\i}sa\v{t} vlastnosti integrovate\v{l}n\'{a}ch funkci\'{\i},
\item vysvetli\v{t} pojem element\'{a}rnej oblasti v $\mathbf{R}^{2},\,%
\mathbf{R}^{3},$
\item aplikova\v{t} vety o v\'{y}po\v{c}te integr\'{a}lov funkci\'{\i} viacer%
\'{y}ch premenn\'{y}ch,
\item aplikova\v{t} pri v\'{y}po\v{c}te integr\'{a}lov vety o substit\'{u}%
cii pomocou pol\'{a}rnych, cylindrick\'{y}ch a sf\'{e}rick\'{y}ch s\'{u}radn%
\'{\i}c.
\end{itemize}
\paragraph{Po\v{z}adovan\'{e} vedomosti:}
\begin{itemize}
\item znalos\v{t} pavidiel a v\'{y}po\v{c}tov\'{y}ch techn\'{\i}k pre v\'{y}%
po\v{c}et jednorozmern\'{y}ch integr\'{a}lov,
\item znalos\v{t} kvadratick\'{y}ch pl\^{o}ch, ich element\'{a}rnych
vlastnost\'{\i} a grafov.
\end{itemize}
\subsection{Integrovanie cez $n$-rozmern\'{e} kv\'{a}dre.}
V Matematickej anal\'{y}ze I sme pre po \v{c}astiach spojit\'{u} funkciu
funkciu $f:\left\langle a,b\right\rangle \longrightarrow \mathbf{R}$
zaviedli pojem Riemannovho integr\'{a}lu oby\v{c}ajne ozna\v{c}ovan\'{e}ho
ako $\int_{a}^{b}f\left( x\right) dx$. Tento pr\'{\i}stup roz\v{s}\'{\i}rime
na funkcie viacer\'{y}ch premenn\'{y}ch.
Anal\'{o}giou intervalu z $\mathbf{R}$ je pojem pravouholn\'{\i}ka (kv\'{a}%
dra) v $\mathbf{R}^{n}.$
\begin{definition}
$n$-dimenzion\'{a}lnym pravouholn\'{\i}kom ($n$-rozmern\'{y}m kv\'{a}drom) $%
E\in \mathbf{R}^{n}$ budeme naz\'{y}va\v{t} kart\'{e}zsky s\'{u}\v{c}in $n$
intervalov, t.j. ak $\mathbf{a}=(a_{1},a_{2},\dots ,a_{n}),\,\mathbf{b}%
=(b_{1},b_{2},\dots ,b_{n}),$ pri\v{c}om $a_{i}$ $<$ $b_{i},\,\forall
i=1,2,\dots ,n.$
\[
E(\mathbf{a},\mathbf{b})=\left\{ \mathbf{x\in R}^{n};a_{i}\leq x_{i}\leq
b_{i},i=1,2,\dots ,n\right\} .
\]
\end{definition}
V pr\'{\i}pade, \v{z}e $n=1$ je to uzavrer\'{y} interval, ak $n=2$ potom je
to obd\'{l}\v{z}nik, pre $n=3$ kv\'{a}der a vo v\v{s}eobecnom pr\'{\i}pade
je to $n$-kv\'{a}der so stranami d\'{l}\v{z}ky $b_{i}-a_{i}.$
Intervaly maj\'{u} d\'{l}\v{z}ku, obd\'{l}\v{z}niky plochu, kv\'{a}dre
objemy. Aby sme tento pojem mohli pomenova\v{t} v\v{s}eobecnou terminol\'{o}%
giou zavedieme tzv. Jordanovsk\'{y} obsah alebo iba obsah. Preto $n$%
-dimenzion\'{a}lny obsah $n$-kv\'{a}dra $E(\mathbf{a},\mathbf{b})$
definujeme
\[
c(E)=\prod_{i=1}^{n}(b_{i}-a_{i})
\]%
Poznamenajme na tomto mieste, \v{z}e napr\'{\i}klad interval $\left\langle
a,b\right\rangle $ m\^{o}\v{z}e by\v{t} pova\v{z}ovan\'{y} aj za degenerovan%
\'{y} obd\'{l}\v{z}nik $\left\langle a,b\right\rangle \times \left\langle
c,d\right\rangle $ s $c=d.$ Jeho 1-dimenzion\'{a}lny obsah je
\[
c(\left\langle a,b\right\rangle )=b-a,
\]%
ale 2-dimenzion\'{a}lny obsah je
\[
c(\left\langle a,b\right\rangle \times \left\langle c,d\right\rangle
)=(b-a)(d-c)=0.
\]%
\emph{Kone\v{c}n\'{e}} zjednotenie $n$-kv\'{a}drov $E_{i},\;S=%
\bigcup_{i=1}^{n}E_{i}$ sa naz\'{y}va \emph{neprekr\'{y}vaj\'{u}ce,} ak ka%
\v{z}d\'{e} dva kv\'{a}dre nemaj\'{u} spolo\v{c}n\'{e} vn\'{u}torn\'{e} body
$int(E_{i})\cap int(E_{j})=\emptyset ,i\neq j,i,j=1,2,\dots ,n.$ Ak $S$ je
tak\'{e} kone\v{c}n\'{e} zjednotenie, tak
\[
c(S)=\sum_{i=1}^{n}c(E_{i}).
\]%
Ak plat\'{\i} $S=\bigcup_{i=1}^{n}E_{i}$ aj $S=\bigcup_{j=1}^{p}F_{j},$
potom $c(S)$ je tak\'{y} ist\'{y} pre obe kone\v{c}n\'{e} zjednotenia. Ak $%
S_{1},S_{2}$ s\'{u} dve neprekr\'{y}vaj\'{u}ce sa mno\v{z}iny (kone\v{c}n%
\'{e} zjednotenia), tak plat\'{\i}
\[
c(S_{1}\cup S_{2})=c(S_{1})+c(S_{2}).
\]%
T\'{u}to vlastnos\v{t} ihne\v{d} mo\v{z}no roz\v{s}\'{\i}ri\v{t} na ka\v{z}d%
\'{e} kone\v{c}n\'{e} zjednotenie, je to tzv \emph{kone\v{c}n\'{a} adit\'{\i}%
vnos\v{t} obsahu.}
Defin\'{\i}cia Riemannovho integr\'{a}lu re\'{a}lnej funkcie definovanej na $%
n$-kv\'{a}dri je tak\'{a} ist\'{a} ako defin\'{\i}cia integr\'{a}lu re\'{a}%
lnej funkcie na intervale. Hlavn\'{y} rozdiel je v tom, \v{z}e d\'{l}\v{z}ku
intervalu zamen\'{\i}me za obsah $n$-kv\'{a}dra.
\begin{definition}
Delenie $\mathcal{P}$ intervalu $\left\langle a,b\right\rangle $ je kone\v{c}%
n\'{a} mno\v{z}ina $\left\{ x_{0},x_{1},\dots ,x_{p}\right\} $ tak\'{a},
\v{z}e
\[
a=x_{0}