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\def\ce{\mathbb C} 
\def\en{\mathbb N}
\def\zet{\mathbb Z}
\def\er{\mathbb R}
\def\Rr{\mathbb R^*}
\def\C{\mathcal{C}}
\def\R{\er}
\def\N{\en}
\def\nec{Nech\v t\ }
\def\ov{\overline}
\def\Int{\operatorname{Int}}
\def\eqdef{\stackrel{\text{\tiny def}}{=}}
\def\Mod{\operatorname{mod}}
\def\dist{\operatorname{dist}}
\def\lp{\displaystyle\lim_{n\to\infty}}
\newcommand\lx[1][0]{\displaystyle\lim_{x\to #1}}
\def \cotg {\operatorname{cotg}}
\def \tg {\operatorname{tg}}
\def \arctg {\operatorname{arctg}}
\def \suma {\sum_{n=1}^\infty}
\def \sumaO {\sum_{n=0}^\infty}
\def \sumad {\sum_{k=2}^\infty}
\def \dx {\, \mathrm{d} x}
\def\lp{\displaystyle\lim_{n\to\infty}}
\def\setsep{:\;}
\def\dl{\, \mathrm{d}\lambda}

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\title{Cvi\v{c}en\'i - Teorie m\'iry a integr\'alu, MFF}  % tyto dvě položky jsou zde v podstatě formálně, ve skutečnosti nejsou nikde 
\author{Marek C\'uth}

\begin{document} 
\centerline{\MakeUppercase{i. konvergence lebesgueova integr\'alu}}\ \\
\textbf{Ur\v{c}ete, zda Lebesgueovy integr\'aly existuj\'i a zda jsou konvergentn\'i:}\\[5pt] \resetpr
\textbf{1.}	\pr $\int_0^\infty \tfrac{x^2}{x^4+1} \dx$
	\pr $\int_0^1 \tfrac{\log(1+x)}{x} \dx$
	\pr $\int_0^1 \tfrac{1}{\sqrt[3]{x(e^x - e^{-x})}} \dx$
	\pr $\int_0^1 \tfrac{1}{\log x} \dx$
	\pr $\int_0^\infty \tfrac{\arctan x}{x^k} \dx\;\; (k\in\er)$\\
	\pr $\int_1^\infty \sin^2\tfrac 1x \dx$
	\pr $\int_0^1 (\log x)e^{-x^2} \dx$
	\pr $\int_0^{\pi/2} \tfrac{\log(\sin x)}{\sqrt{x}} \dx$
	\pr $\int_{-1}^1 \cos\left(\sin \tfrac 1x + e^{\tfrac 1x}\right) \dx$\\[10pt]\resetpr
\textbf{2. P\v{r}\'iklady s parametrem ($\alpha, \beta\in\er$):} \pr $\int_0^\pi \tfrac{\sin x}{x^\alpha (\pi - x)} \dx$
	\pr $\int_2^\infty \tfrac{(\log x)^\beta}{x^\alpha} \dx$
	\pr $\int_0^{1/2} x^\alpha |\log x|^\beta \dx$\\
	\pr $\int_0^1 \tfrac{\log(1-\alpha^2x^2)}{x^2\sqrt{1-x^2}} \dx$
	\pr $\int_0^\infty \tfrac{x^{\alpha - 1} - x^{\beta - 1}}{1-x} \dx\;\; (\alpha > \beta)$
	\pr $\int_0^\pi \log(\sin \alpha x) \dx$\\
	\pr $\int_0^{\pi/2} (\tan x)^\alpha \dx$
	\pr $\int_0^1 x^{\alpha x} \dx$
	\pr $\int_0^\infty (\pi - 2\arctan x)^\alpha \dx$
	\pr $\int_0^1 \tfrac{\arccos x}{\log^\alpha(1/x)} \dx$\\[20 pt]
\centerline{\MakeUppercase{ii. integrace posloupnost\'i a \v{r}ad funkc\'i}}\ \\[5pt]
\textbf{1. Spo\v{c}t\v{e}te n\'asleduj\'ic\'i limity posloupnost\'i: }\resetpr \pr $\lp \int_0^1 \frac{n^{3/2}x}{1 + n^2x^2}$
	\pr $\lp \int_0^\infty \frac{x^n}{1 + x^{2n}} \dx$\\
	\pr $\lp \int_0^\infty \frac{1}{\left(1+\tfrac{x}{n}\right)^n\sqrt[n]{x}} \dx$
	\pr $\lp \int_0^\infty \frac{\log(x+n)}{n}e^{-x} \dx$
	\pr $\lp \int_0^{\pi/4} \frac{1}{\log\left(\tfrac{n-1}{n} - \sin x\right)} \dx$\\[10pt]
\textbf{2. Spo\v{c}t\v{e}te n\'asleduj\'ic\'i limity:}\resetpr \pr $\displaystyle\lim_{a\to \infty}\int_0^\infty e^{-ax^2} \dx$
	\pr $\displaystyle\lim_{a\to 0^+}\int_0^\infty e^{-ax^2} \dx$\\
	\pr $\displaystyle\lim_{a\to 1^-}\int_0^{\pi/4} \frac{1}{\log(a - \sin x)} \dx$
	\pr $\displaystyle\lim_{a\to \infty}\int_0^\infty \frac{e^{-ax}}{1 + x^2} \dx$
	\pr $\displaystyle\lim_{a\to \infty}\int_0^\infty e^{-ax}\sin x \dx$\\[10pt]
\textbf{3. V n\'asleduj\'ic\'ich p\v{r}\'ikladech rozvi\v{n}te integrovanou funkci v \v{r}adu, ov\v{e}\v{r}te mo\v{z}nost z\'am\v{e}ny \v{r}ady a integr\'alu a vyj\'ad\v{r}ete integr\'al jako \v{c}\'iselnou \v{r}adu.}\\[5pt] \resetpr
	\pr $\int_0^1 \tfrac{\log(1-x)}{x} \dx$
	\pr $\int_0^1 \tfrac{\log(1+x)}{x} \dx$
	\pr $\int_0^\infty \tfrac{x}{e^x-1} \dx$
	\pr $\int_0^\infty \tfrac{x}{e^x+1} \dx$
	\pr $\int_0^1 \tfrac{x^{p-1}}{1 + x^q} \dx\;\; (p,q > 0)$\\
	\pr $\int_0^\infty e^{-x}\cos\sqrt{x} \dx$
	\pr $\int_0^\infty \log x \log(1-x) \dx$
	\pr $\int_0^\infty \log x \log(1+x) \dx$	
	\pr $\int_0^\infty \tfrac{\sin ax}{e^x - 1} \dx\;\; (a\in\er)$\\
	\pr $\int_0^1 \tfrac{|\log x|}{1-x^2} \dx$
	\pr $\int_0^1 \tfrac{x^2|\log x|}{1+x}$
	\pr $\int_0^\infty \log(1-e^{-x}) \dx$
	\pr $\int_0^1 \tfrac{e^x-1}{x} \dx$
	\pr $\int_0^1 \tfrac{1+x}{1-x}|\log x| \dx$
	\pr $\int_0^\infty \tfrac{1}{e^{8x} + 1} \dx$\\[20pt]
\centerline{\MakeUppercase{iii. integr\'aly z\'avisl\'e na parametru}}\ \\[5pt]
\textbf{1. Ur\v{c}ete defini\v{c}n\'i obor n\'asleduj\'ic\'ich re\'aln\'ych funkc\'i (= mno\v{z}inu v\v{s}ech $a\in\er$, \v{z}e $F(a)\in\er$) a doka\v{z}te, \v{z}e jsou na sv\'em defini\v{c}n\' im oboru spojit\'e.}\\[5pt] \resetpr
	\pr $F(a) = \int_0^\infty \tfrac{e^{-ax}}{1+x^2} \dx$
	\pr $F(a) = \int_0^\infty e^{-ax} \dx$
	\pr $F(a) = \int_{-1}^1 \sqrt{x^2 + a^2} \dx$
	\pr $F(a) = \int_0^1 \tfrac{1}{\sqrt{x^2 + a^2}} \dx$\\
	\pr $F(a) = \int_0^\infty \tfrac{x}{2 + x^a} \dx$
	\pr $F(a) = \int_{1/2}^\infty \tfrac{\cos x}{x^a} \dx$
	\pr $F(a) = \int_0^\infty x^{a-1}e^{-x} \dx$
	\pr $F(a) = \int_0^\infty \tfrac{x^{a-1}}{1+x} \dx$\\
	\pr $F(a) = \int_1^\infty \tfrac{\sin 1/x}{x(x+a)^2} \dx$
	\pr $F(a) = \int_0^1 \tfrac{x^a}{\sqrt{1-x^2}} \dx$
	\pr $F(a) = \int_0^1 \log(x^2 + a^2) \dx$
	\pr $F(a) = \int_0^\infty \tfrac{\sin x}{x}e^{-ax} \dx$\\[10pt]
\textbf{2. Rozhodn\v{e}te, pro kter\'e hodnoty parametr\r{u} konverguj\'i n\'asleduj\'ic\'i Lebesgueovy
integr\'aly a spo\v{c}t\v{e}te jejich hodnoty pomoc\'i v\v{e}ty o derivov\'an\'i integr\'al\r{u}
z\'avisl\'ych na parametru.}\\[5pt] \resetpr
	\pr $\displaystyle\int_0^{\pi/2} \frac{\arctg(a\tg x)}{\tg x} \dx$
	\pr $\displaystyle\int_0^\infty \frac{1 - e^{-ax}}{xe^x} \dx$
	\pr $\displaystyle\int_0^\infty \frac{1 - e^{-ax^2}}{xe^{x^2}} \dx$
	\pr $\displaystyle\int_0^1 \frac{1-x^a}{\log x} \dx$\\
	\pr $\displaystyle\int_0^\infty e^{-ax}\frac{\sin bx}{x} \dx$
	\pr $\displaystyle\int_0^\infty \frac{e^{-ax^2} - e^{-bx^2}}{x} \dx$
	\pr $\displaystyle\int_0^\infty \frac{\arctg(ax) - \arctg(bx)}{x} \dx$\\
	\pr $\displaystyle\int_0^\pi \frac{\log(1+a\cos x)}{\cos x} \dx$
	\pr $\displaystyle\int_0^1 \frac{\log(1 - a^2x^2)}{\sqrt{1-x^2}} \dx$
	\pr $\displaystyle\int_0^{\pi/2} \frac{\arctg(a\sin x)}{\sin x} \dx$
	\pr $\displaystyle\int_0^1 \frac{\arctg(ax)}{x\sqrt{1-x^2}} \dx$\\
	\pr $\displaystyle\int_0^\infty \frac{\log(a^2 + x^2)}{b^2 + x^2} \dx$
	\pr $\displaystyle\int_0^\infty e^{-ax} \frac{1 - \cos x}{x} \dx$\\[20pt]
\centerline{\MakeUppercase{iv. v\'icerozm\v{e}rn\'a integrace}}\ \\[5pt]
\textbf{1. Spo\v{c}t\v{e}te m\'iru $\lambda_2(M)$ mno\v{z}iny $M$ (nepou\v{z}\'ivejte v\v{e}tu o substituci)}\\[5pt] \resetpr
\pr $M = \{[x,y]\in\er^2\setsep 1\leq x \leq5,\;y\leq 3,\;xy\geq 1\}$
\pr $M = \{[x,y]\in\er^2\setsep x>2,\;0 < y < 1/x^2\}$\\
\pr $M = \{[x,y]\in\er^2\setsep x>2,\;0 < y < 1/x\}$\\
\pr $M$ je omezen\'a k\v{r}ivkami: $2x-y = 0,\quad 2x-y-7=0,\quad x-4y+7=0,\quad x-4y+14=0$\\
\pr $M$ je omezen\'a k\v{r}ivkami: $x = \tfrac{y^2+b^2}{2b},\quad x = \tfrac{y^2+a^2}{2a}\quad(0<b<a)$\\[10pt]
\textbf{2. Spo\v{c}t\v{e}te $\int\int_{M}f(x,y) \dl^2$ (nepou\v{z}\'ivejte v\v{e}tu o substituci)}\\[5pt] \resetpr
\pr $M = [0,1]\times [0,1]$;\qquad $f(x,y) = \tfrac{x^2}{1+y^2}$\\
\pr $M$ je ohrani\v{c}en\'a k\v{r}ivkami $y = 0$, $y = 1 - x$, $y = 1 + x$;\qquad $f(x,y) = x^2 + y^2$\\
\pr $M$ je ohrani\v{c}en\'a k\v{r}ivkami $y = x^2$, $y^2 = x$;\qquad $f(x,y) = x^2 + y$\\
\pr $M = \{[x,y]\in\er^2\setsep x^2 + y^2\leq R^2\}$\quad($R > 0$);\qquad $f(x,y) = y^2\sqrt{R^2 - x^2}$\\
\pr $M = \{[x,y]\in\er^2\setsep x^2 + y^2\leq x\}$;\qquad $f(x,y) = \tfrac{1}{\sqrt{y^2 + x^2}}$\\
\pr $M$ je ohrani\v{c}en\'a k\v{r}ivkami $y = 0$, $x = 1$, $y = x$;\qquad $f(x,y) = \sqrt{4x^2 - y^2}$\\[10pt]
\textbf{3. Spo\v{c}t\v{e}te $\int\int_{M}f(x,y) \dl^2$}\\[5pt] \resetpr
\pr $M = \{[x,y]\in\er^2\setsep x^2 + y^2\leq 1\}$;\qquad $f(x,y) = \tfrac{1}{\sqrt{1-x^2-y^2}}$\\
\pr $M = \{[x,y]\in\er^2\setsep x^2 + y^2\leq x\}$;\qquad $f(x,y) = \tfrac{1}{\sqrt{y^2 + x^2}}$\\
\pr $M = \{[x,y]\in\er^2\setsep \left(\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2}\right)^2\leq\tfrac{x^2y}{c^3},\;x\geq 0,\;y\geq 0\}$\quad($a,b,c>0$); \qquad $f(x,y) = xy$\\
\pr $M = \{[x,y]\in\er^2\setsep \sqrt{x} + \sqrt{y}\leq 1,\;x\geq 0,\;y\geq 0\}$;\qquad $f(x,y) = \sqrt{\sqrt{x} + \sqrt{y}}$\\
\pr $M = \{[x,y]\in\er^2\setsep \left(\tfrac{(x+2)^2}{2} + \tfrac{y^2}{3}\right)^4\leq\tfrac{(x+2)y}{6}\}$; \qquad $f(x,y) = \sqrt{xy + 2y}$\\[10pt]
\textbf{4. Spo\v{c}t\v{e}te m\'iru $\lambda_2(M)$ mno\v{z}iny $M$} (v \'uloh\'ach uva\v{z}ujte v\v{s}echny parametry kladn\'e)\\[5pt] \resetpr
\pr $M = \{[x,y]\in\er^2\setsep (x^2 + y^2)^2\leq 2a^2(x^2 - y^2)\}$\quad(obsah plochy ohrani\v{c}en\'e lemnisk\'atou)\\
\pr $M = \{[x,y]\in\er^2\setsep (x^2 + y^2)^3 < a^2(x^4 + y^4)\}$\\
\pr $M = \{[x,y]\in\er^2\setsep x^2 + y^2\leq R^2,\;x^2 + y^2\leq 2Ry,\;x\geq 0\}$\\
\pr $M = \{[x,y]\in\er^2\setsep 2 < x+y < 3,\;x < y < 3x\}$\qquad
\pr $M = \{[x,y]\in\er^2\setsep (x+y)^3 < axy,\;x>0,\;y>0\}$\\
\pr $M = \{[x,y]\in\er^2\setsep a < xy < b,\;py < x < qy$\quad($0<a<b$, $0<p<q$)\\
\pr $M$ je ohrani\v{c}en\'a k\v{r}ivkou $\left(\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2}\right)^2 = x^2 + y^2$\\[10pt]
\textbf{5. Pomoc\'i Fubiniovy v\v{e}ty spo\v{c}t\v{e}te hodnoty n\'asleduj\'ic\'ich Lebesgueov\'ych integr\'al\r{u}}\\
(m\r{u}\v{z}ete pou\v{z}\'ivat tzv. Laplace\r{u}v integ\'al, tj. $\int_0^\infty e^{-x^2} \dx = \sqrt{\pi}/2$)\\[5pt] \resetpr
\pr $\displaystyle\int_0^\infty \frac{\arctg(ax) - \arctg(bx)}{x} \dx$\quad($ab > 0$)
\pr $\displaystyle\int_0^\infty \frac{e^{-ax^2} - e^{-bx^2}}{x^2} \dx$\quad($a\geq0$,\;$b\geq0$)\\
\pr $\displaystyle\int_0^\infty \frac{e^{-ax^2} - e^{-bx^2}}{x} \dx$\quad($a>0$,\;$b>0$)
\pr $\displaystyle\int_0^\infty \frac{e^{-ax} - e^{-bx}}{x} \dx$\quad($a>0$,\;$b>0$)\\
\pr $\displaystyle\int_0^\infty \frac{\log(1+a^2x^2) - \log(1+b^2x^2)}{x^2} \dx$\quad($a,b\in\er$)
\pr $\displaystyle\int_0^\infty (e^{-a^2/x^2} - e^{-b^2/x^2}) \dx$\quad($a,b\in\er$)\\
\pr $\displaystyle\int_0^1 \frac{x^b - x^a}{\log x} \dx$\quad($a>-1,\;b>-1$)\\[10pt]
\textbf{6. Spo\v{c}t\v{e}te m\'iru $\lambda_3(M)$ mno\v{z}iny $M$}(v \'uloh\'ach uva\v{z}ujte v\v{s}echny parametry kladn\'e) \\[5pt] \resetpr
\pr $M = \{[x,y,z]\in\er^3\setsep x^2 + y^2 + z^2\leq 2az,\;x^2 + y^2\leq z^2\}$\\
\pr $M = \{[x,y,z]\in\er^3\setsep (x + y + z)^2 < ay,\;x > 0,\;y > 0,\;z > 0\}$\\
\pr $M = \{[x,y,z]\in\er^3\setsep x^2 + y^2 + z^2\leq R^2,\;x^2 + y^2\leq Rx\}$\quad(objem Vivianiho ok\'enka)\\
\pr $M = \{[x,y,z]\in\er^3\setsep (R - \sqrt{x^2 + y^2})^2 + z^2\leq a^2\}$, $R>a$\quad(objem anuloidu)\\
\pr $M = \{[x,y,z]\in\er^3\setsep \left(\tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} + \tfrac{z^2}{c^2}\right)^2\leq \tfrac{x^2y}{h^3}\}$\\
\pr $M = \{[x,y,z]\in\er^3\setsep c(x^2 + y^2) + a^2z\leq a^2c,\;z\geq 0\}$\\
\pr $M = \{[x,y,z]\in\er^3\setsep \tfrac{y^2}{2} + \tfrac{z^2}{3}\leq \tfrac{1}{x^2+2x+3}\}$\quad (p\v{r}\'iklad ze zkou\v{s}kov\'e p\'isemky v akademick\'em roce 2009/10)\\[10pt]
\textbf{7. Spo\v{c}t\v{e}te $\int\int\int_{M}f(x,y,z) \dl^3$}\quad(v \'uloh\'ach uva\v{z}ujte v\v{s}echny parametry kladn\'e)\\[5pt] \resetpr
\pr $M = \{[x,y,z]\in\er^3\setsep x^2 + y^2 + z^2\leq R^2,\;x^2 + y^2 + z^2\leq 2Rz\}$;\qquad$f(x,y,z) = z^2$\\
\pr $M = \{[x,y,z]\in\er^3\setsep \tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} + \tfrac{z^2}{c^2}\leq 1\}$;\qquad$f(x,y,z) = \tfrac{x^2}{a^2} + \tfrac{y^2}{b^2} + \tfrac{z^2}{c^2}$\\
\pr $M = \{[x,y,z]\in\er^3\setsep x^2 + y^2 \leq 3z,\;x^2 + y^2 + z^2\leq 4\;\}$;\qquad$f(x,y,z) = z$\\
\pr $M = \{[x,y,z]\in\er^3\setsep x^2 + y^2 + z^2\leq z\;\}$;\qquad$f(x,y,z) = \sqrt{x^2 + y^2 + z^2}$\\
\end{document}