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Revision difference : integral over plane region |
| Version 9 |
Version 8 |
| \PMlinkescapeword{side} \PMlinkescapeword{inner} \PMlinkescapeword{right} |
\PMlinkescapeword{side} \PMlinkescapeword{inner} \PMlinkescapeword{right} |
| The integrals over a planar region are generalisations of usual Riemann integrals, but special cases of \PMlinkid{surface integrals}{6660}. |
The integrals over a planar region are generalisations of usual Riemann integrals, but special cases of \PMlinkid{surface integrals}{6660}. |
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| \subsection{Integral over a rectangle} |
\subsection{Integral over a rectangle} |
| Let $R$ be the rectangle of $xy$-plane defined by |
Let $R$ be the rectangle of $xy$-plane defined by |
| \begin{align} |
\begin{align} |
| a \leqq x \leqq b, \quad c \leqq y \leqq d |
a \leqq x \leqq b, \quad c \leqq y \leqq d |
| \end{align} |
\end{align} |
| and the function $f$ be defined and bounded in $R$.\, Let |
and the function $f$ be defined and bounded in $R$.\, Let |
| \begin{align} |
\begin{align} |
| D: |
D: |
| \begin{cases} |
\begin{cases} |
| x_0 = a,\,x_1,\,\ldots,\,x_m = b\\ |
x_0 = a,\,x_1,\,\ldots,\,x_m = b\\ |
| y_0 = c,\,y_1,\,\ldots,\,y_n = d |
y_0 = c,\,y_1,\,\ldots,\,y_n = d |
| \end{cases} |
\end{cases} |
| \end{align} |
\end{align} |
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a \PMlinkescapetext{division} of $R$ into the rectangular parts $\Delta_i$ with areas $\Delta_iA$ ($i = 1,\,\ldots,\,mn$).\, Denote
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a \PMlinkescapetext{division} of $R$ into the parts $\Delta_i$ with areas $\Delta_iA$ ($i = 1,\,\ldots,\,mn$).\, Denote
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| $$m_i \;:=\; \inf_{\Delta_i}f(x,\,y), \qquad M_i \;:=\; \sup_{\Delta_i}f(x,\,y)$$ |
$$m_i \;:=\; \inf_{\Delta_i}f(x,\,y), \qquad M_i \;:=\; \sup_{\Delta_i}f(x,\,y)$$ |
| and |
and |
| $$s_D \;:=\; \sum_Dm_i\Delta_iA, \qquad S_D \;:=\; \sum_DM_i\Delta_iA.$$\\ |
$$s_D \;:=\; \sum_Dm_i\Delta_iA, \qquad S_D \;:=\; \sum_DM_i\Delta_iA.$$\\ |
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| \textbf{Definition 1.}\; If\; $\displaystyle\sup_D\{s_D\} \,=\, \inf_D\{S_D\}$,\, then we say that $f$ is \emph{integrable over} $R$ and call the common value the (\emph{Riemann}) \emph{integral of $f$ over the rectangle} $R$ and denote it by |
\textbf{Definition 1.}\; If\; $\displaystyle\sup_D\{s_D\} \,=\, \inf_D\{S_D\}$,\, then we say that $f$ is \emph{integrable over} $R$ and call the common value the (\emph{Riemann}) \emph{integral of $f$ over the rectangle} $R$ and denote it by |
| $$\int_Rf, \quad \int_Rf(x,\,y)\,dx\,dy \quad \mbox{or} \quad \iint_Rf(x,\,y)\,dx\,dy.$$\\ |
$$\int_Rf, \quad \int_Rf(x,\,y)\,dx\,dy \quad \mbox{or} \quad \iint_Rf(x,\,y)\,dx\,dy.$$\\ |
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| Let then $f$ be defined in a region $A$ of $xy$-plane such that that it can be enclosed in a rectangle $R$ defined by (1).\, Define the new function $f_1$ through |
Let then $f$ be defined in a region $A$ of $xy$-plane such that that it can be enclosed in a rectangle $R$ defined by (1).\, Define the new function $f_1$ through |
| \begin{align} |
\begin{align} |
| f_1(x,\,y) \;:=\; |
f_1(x,\,y) \;:=\; |
| \begin{cases} |
\begin{cases} |
| f(x,\,y) \quad\mbox{when\;\;} (x,\,y) \in A,\\ |
f(x,\,y) \quad\mbox{when\;\;} (x,\,y) \in A,\\ |
| 0 \qquad\mbox{otherwise.} |
0 \qquad\mbox{otherwise.} |
| \end{cases} |
\end{cases} |
| \end{align} |
\end{align} |
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| \textbf{Definition 2.}\; If $f_1$ is integrable over the rectangle $R$, we say that $f$ is \emph{integrable over} $A$ and define |
\textbf{Definition 2.}\; If $f_1$ is integrable over the rectangle $R$, we say that $f$ is \emph{integrable over} $A$ and define |
| \begin{align} |
\begin{align} |
| \int_Af \;:=\; \int_Rf_1. |
\int_Af \;:=\; \int_Rf_1. |
| \end{align} |
\end{align} |
| It's apparent that (4) is \PMlinkescapetext{independent} on the choice of $R$ since the points of $\mathbb{R}^2\!\smallsetminus\!A$ give zero-terms to the lower and upper sums. |
It's apparent that (4) is \PMlinkescapetext{independent} on the choice of $R$ since the points of $\mathbb{R}^2\!\smallsetminus\!A$ give zero-terms to the lower and upper sums. |
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| \subsection{Double integrals} |
\subsection{Double integrals} |
| \textbf{Definition 3.}\; Let $f$ be bounded in $R$ as before.\, Suppose that |
\textbf{Definition 3.}\; Let $f$ be bounded in $R$ as before.\, Suppose that |
| $$\varphi(x) \;:=\; \int_c^df(x,\,y)\,dy$$ |
$$\varphi(x) \;:=\; \int_c^df(x,\,y)\,dy$$ |
| is defined on\, $[a,\,b]$.\, If also the integral |
is defined on\, $[a,\,b]$.\, If also the integral |
| \begin{align} |
\begin{align} |
| \int_a^b\varphi(x)\,dx \;=\; \int_a^b\left[\int_c^df(x,\,y)\,dy\right]dx |
\int_a^b\varphi(x)\,dx \;=\; \int_a^b\left[\int_c^df(x,\,y)\,dy\right]dx |
| \end{align} |
\end{align} |
| exists, it is called a \emph{double integral} or \emph{iterated integral} and denoted by |
exists, it is called a \emph{double integral} or \emph{iterated integral} and denoted by |
| $$\int_a^bdx\int_c^df(x,\,y)\,dy.$$\\ |
$$\int_a^bdx\int_c^df(x,\,y)\,dy.$$\\ |
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| One may prove the |
One may prove the |
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| \textbf{Theorem.}\, $\displaystyle\int_Rf(x,\,y)\,dx\,dy \,=\, \int_a^bdx\int_c^df(x,\,y)\,dy$,\; provided that the integral of the left side exists and that the inner integral $\int_c^df(x,\,y)\,dy$ of the right side exists for every $x$ in\, $[a,\,b]$.\\ |
\textbf{Theorem.}\, $\displaystyle\int_Rf(x,\,y)\,dx\,dy \,=\, \int_a^bdx\int_c^df(x,\,y)\,dy$,\; provided that the integral of the left side exists and that the inner integral $\int_c^df(x,\,y)\,dy$ of the right side exists for every $x$ in\, $[a,\,b]$.\\ |
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| It's clear that\; $\displaystyle\int_a^bdx\int_c^df(x,\,y)\,dy \,=\, \int_c^ddy\int_a^bf(x,\,y)\,dx$\; if also the integral $\int_a^bf(x,\,y)\,dx$ exists for every $y$ in\, $[c,\,d]$.\, If especially the function $f$ is continuous in the rectangle $R$, then surely |
It's clear that\; $\displaystyle\int_a^bdx\int_c^df(x,\,y)\,dy \,=\, \int_c^ddy\int_a^bf(x,\,y)\,dx$\; if also the integral $\int_a^bf(x,\,y)\,dx$ exists for every $y$ in\, $[c,\,d]$.\, If especially the function $f$ is continuous in the rectangle $R$, then surely |
| $$\int_Rf(x,\,y)\,dx\,dy \;=\; \int_a^bdx\int_c^df(x,\,y)\,dy \;=\; \int_c^ddy\int_a^bf(x,\,y)\,dx.$$\\ |
$$\int_Rf(x,\,y)\,dx\,dy \;=\; \int_a^bdx\int_c^df(x,\,y)\,dy \;=\; \int_c^ddy\int_a^bf(x,\,y)\,dx.$$\\ |
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| Assume now, that $f$ is defined and bounded in the region |
Assume now, that $f$ is defined and bounded in the region |
| $$A \;:=\; \{(x,\,y) \in \mathbb{R}^2\vdots\; a \leqq x \leqq b,\; h_1(x) \leqq y \leqq h_2(x)\}$$ |
$$A \;:=\; \{(x,\,y) \in \mathbb{R}^2\vdots\; a \leqq x \leqq b,\; h_1(x) \leqq y \leqq h_2(x)\}$$ |
| where $A$ is contained in the rectangle $R$ determined by (1).\, Then the planar integral $\displaystyle\int_Af$ is defined as |
where $A$ is contained in the rectangle $R$ determined by (1).\, Then the planar integral $\displaystyle\int_Af$ is defined as |
| $$\int_Af \;=\; \int_Rf$$ |
$$\int_Af \;=\; \int_Rf$$ |
| if the integral of right side exists.\, For this, he continuity of $f$ in $A$ does not necessarily suffice, because $f_1$ may have a jump discontinuity on the border of $A$ whence the integrability of $f_1$ needs not be guaranteed.\, One case where the integrability is true is that the graphs of the functions $h_1$ and $h_2$ are rectifiable (i.e. the functions have continuous derivatives).\, For a continuous $f$, we then have |
if the integral of right side exists.\, For this, he continuity of $f$ in $A$ does not necessarily suffice, because $f_1$ may have a jump discontinuity on the border of $A$ whence the integrability of $f_1$ needs not be guaranteed.\, One case where the integrability is true is that the graphs of the functions $h_1$ and $h_2$ are rectifiable (i.e. the functions have continuous derivatives).\, For a continuous $f$, we then have |
| $$\int_c^df_1(x,\,y)\,dy \;=\; \int_c^{h_1(x)}0\,dy +\int_{h_1(x)}^{h_2(x)}f(x,\,y)\,dy+\int_{h_2(x)}^d0\,dy |
$$\int_c^df_1(x,\,y)\,dy \;=\; \int_c^{h_1(x)}0\,dy +\int_{h_1(x)}^{h_2(x)}f(x,\,y)\,dy+\int_{h_2(x)}^d0\,dy |
| \;=\; \int_{h_1(x)}^{h_2(x)}f(x,\,y)\,dy.$$ |
\;=\; \int_{h_1(x)}^{h_2(x)}f(x,\,y)\,dy.$$ |
| Thus |
Thus |
| \begin{align} |
\begin{align} |
| \int_Af \;=\; \int_Rf_1 \;=\; \int_a^bdx\int_{h_1(x)}^{h_2(x)}f(x,\,y)\,dy, |
\int_Af \;=\; \int_Rf_1 \;=\; \int_a^bdx\int_{h_1(x)}^{h_2(x)}f(x,\,y)\,dy, |
| \end{align} |
\end{align} |
| i.e. the planar integral has been expressed as a double integral. |
i.e. the planar integral has been expressed as a double integral. |
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| [Not ready ...] |
[Not ready ...] |
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