\subsubsection {}
Uka�te, �e existuje otev�en� omezen� i uzav�en� omezen� mno�ina 
v ${\Bbb R }$, kter� nen� m��iteln� 



\subsubsection {}
Doka�te, �e ka�d� izometrick� zobrazen� v ${\Bbb R }^2$ je 
slo�en�m ortogon�ln�ho zobrazen� a translace. Odvo�te, �e zachov�v� 
m��itelnost i m�ru.
 



\subsubsection {}
Doka�te, �e obraz mno�iny m�ry nula p�i lipschitzovsk�m 
zobrazen� m� m�ru nula.

 



\subsubsection {}
Doka�te, �e v�po�et plochy (objemu) mezi dv�ma grafy funkc�, 
kter� jsme pou��vali, d�v� opravdu m�ru p��slu�n� mno�iny.
 



\subsubsection {}
Doka�te, �e m�ry mno�in $A_{n}$ konverguj� k m��e mno�iny $A$, 
jestli�e jde o m��iteln� otev�en� mno�iny, $A_{n+1}\supset A_n$ a 
$\bigcup\limits_{n\in{\Bbb N }} A_n = A$. Je-li $f$ integrovateln� na $A$, 
je $\lim\limits_{N\to\infty} \int\limits{A_n} f(x)\ dx = \int\limits_{A} 
f(x)\ dx.$

 



\subsubsection {}
Spo�t�te objem rovnob�n�ku (rovnob�nost�nu) zadan�ho 
vektory $v_{1}, v_{2}$ v ${\Bbb R }^2$ ($v_{1}, v_{2}, v_{3}$ v ${\Bbb R }^3$) 
pomoc� vhodn�ho 
determinantu. Co je plocha rovnob�n�ku dan�ho dv�ma vektory v $\Bbb 
R^3$? Jak se zm�n� m�ra mno�iny p�i line�rn�m (afinn�m) zobrazen�?

 



\subsubsection {}
 Spo�t�te integr�ly:

$\int\limits_A x^2 + y^2\ dx\ dy$, kde $A = \{(x,y)\mid x^2 + y^2 \le 
2x\}.$

$\int\limits_A \sqrt{1 - x^2 - y^2}\ dx\ dy$, kde $A = \{(x,y)\mid x^2 + 
y^2 \le 1, x\ge 0, y \ge 0\}$.

$\int\limits_A \sqrt{1 - x^2 - y^2}\ dx\ dy$, kde $A = \{(x,y)\mid x^2 + 
y^2 \le 1, x\ge 0, y \ge 0\}$.   

$\int\limits_A \sqrt{1 - x^2 - y^2}\ dx\ dy$, kde $A$ je omezen� mno�ina 
s hranic� $\{(x,y)\mid (x^2 + y^2)^2 = x^2 - y^2\}$.   
 
$\int\limits_A z\ dx\ dy\ dz$, kde $A = \{(x,y,z)\mid \frac{x^2}{a^2} + 
\frac{y^2}{a^2}  + \frac{z^2}{a^2} \le 1\}$.

$\int\limits_A \sqrt{x^2 + y^2 + z^2}\ dx\ dy\ dz$, kde $A = 
\{(x,y,z)\mid x^2 + y^2 + z^2 \le x\}.$

$\int\limits_A x\ dx\ dy\ dz$, kde $A = 
\{(x,y,z)\mid x^2 + y^2 + z^2 \le 4, x\ge 0, y\ge 0, z\ge 0\}.$


 



\subsubsection {}
Spo�t�te m�ru mno�iny

$A = \{(x,y)\mid y\ge x^2, y-x\le 2\}$,

$B = \{(x,y)\mid x^2 + y^2 < 1, x + y \le 1\}$,

$C = \{(x,y)\mid y \ge 1, y < 1 + \cos x, -\frac{\pi}{2} < x < 
\frac{\pi}{2}\}$. 



\subsubsection {}
Spo�t�te m�ru omezen� otev�en� mno�iny, jej� hranice je obsa�ena ve 
sjednocen� ploch definovan�ch rovnicemi:

$$z = y^2, x^2 + y^2 = 1, z = 0;$$

$$y = x^2, z = x^2 + y^2, y = 1, z = 0;$$

$$x + y + z = 2, 3x + y = 2, \frac{3}{2} x + y = 2, y = 0, z = 0;$$

$$x^2 + y^2 + z^2 = r^2, x^2 + y^2 = rx, r > 0;$$

$$x^2 + z^2 = a^2, x^2 + y^2 = a^2, a > 0;$$

$$x^2 + y^2 = a^2, y^2 = bz, z = 0, a,b >0.$$
 



\subsubsection {}
Spo�t�te m�ru otev�en� omezen� mno�iny $A$, jej� hranice je pokryta 
��stmi mno�in $\{(x,y,z)\mid x^2 + y^2 + z^2 = 2z\}$ a $\{(x,y,z)\mid 
x^2 + y^2 = z^2\}$, kter� obsahuje bod $(0,0,1)$;
 



\subsubsection {}
Kolik piva (a kam) vyte�e z pln�ho p�llitru, jestli�e ho napln�n� 
velkopopovick�m le��kem naklon�me tak, �e hladina proch�z� st�edem 
z�kladny? Ov��te. 



\subsubsection {}
Po��tejte nejen plochy a objemy, ale i obvody, t�i�t� r�zn�ch 
rovinn�ch mno�in i t�les v ${\Bbb R }^3$. Po��tejte t� jejich momenty 
setrva�nosti vzhledem k dan�mu "st�edu ot��en�", kinetickou energii p�i 
konstant� hustot� hmotnosti, p��p. konstantn� �hlov� rychlosti. 

Uva�ujte nap�. disk, kolo (mezikru��), kruhovou �se� (setrva�n�k), kouli 
ap.
 
