1103124404
Level:
C
Let a solid be obtained by rotating the blue triangle about the $x$-axis (see the picture). Find such a value of $a$, that the volume of this solid is $48\pi$.
$a=4$
$a=2$
$a=3$
$a=6$
Applications of definite integral
1103124405
Level:
C
Let a solid be obtained by rotating the blue triangle about the $y$-axis (see the picture). Find the volume of this solid.
$32\pi$
$32$
$96\pi$
$100$
2010012603
Level:
C
The instantaneous velocity of a moving body is proportional to the cube of the time. The velocity at the
time \(t = 3\, \mathrm{s}\)
is \(v = 9\, \mathrm{m\, s}^{-1}\).
What is the distance traveled by the body in the first \(6\) seconds?
\(108\, \mathrm{m}\)
\(54\, \mathrm{m}\)
\(324\, \mathrm{m}\)
2010012604
Level:
C
The gravitational force of the attraction of two particles is
\[
F(x) = \frac{c}
{x^{2}},
\]
where \(x\) is the distance
in meters and \(c\)
a positive constant. Find the work required to increase the distance between the particles
from \(2\, \mathrm{m}\)
to \(5\, \mathrm{m}\).
\(\frac{3}
{10}c\, \mathrm{J}\)
\(\frac{2}
{5}c\, \mathrm{J}\)
\(c\, \mathrm{J}\)
2010014301
Level:
C
Find the area of the sickle bounded by half the ellipse and half the circle (see the picture). Points \(A\) and \(B\) lying on the circle are the foci of the ellipse.
\(2\pi(\sqrt{2}-1)\)
\(1.3\)
\(6\pi\)
\(5.2\)
2010014302
Level:
C
Find the area of the sickle bounded by half the ellipse and half the circle (see the picture). Points \(A\) and \(B\) lying on the circle are the foci of the ellipse.
\(\frac{\pi}2(\sqrt{2}-1)\)
\(1.3\)
\(1.5\pi\)
\(0.3\)
2010014303
Level:
C
Find the area of the “smile” bounded by half the circle and half the ellipse (see the picture). Points \(A\) and \(B\) are the foci of the ellipse.
\(\frac{3}2\pi(3-\sqrt{5})\)
\(1.8\)
\(\frac32\pi\)
\(7.2\)
2010014304
Level:
C
Find the area of the “smile” bounded by half the circle and half the ellipse (see the picture). Points \(A\) and \(B\) are the foci of the ellipse.
\(\pi(\sqrt{13}-2)\)
\(2.52\)
\(\pi\)
\(10.09\)
2010014305
Level:
C
Approximately, the shape of the Earth is an ellipsoid. This ellipsoid can be obtained by rotating an ellipse with semi-axes \(a=6\,378\,137\,\mathrm{m}\) and \(b=6\,356\,752\,\mathrm{m}\) around its minor axis. What is the volume \(V\) of this ellipsoid?
\(V\doteq 1.083\cdot 10^{21}\,\mathrm{m}^3 \)
\(V\doteq 1.080\cdot 10^{21}\,\mathrm{m}^3 \)
\(V\doteq 4.002\cdot 10^{14}\,\mathrm{m}^3 \)
\(V\doteq 1.274\cdot 10^{14}\,\mathrm{m}^3 \)
2010014306
Level:
C
Approximately, the shape of Mars is an ellipsoid. This ellipsoid can be obtained by rotating an ellipse with semi-axes \(a=3\,396\,190\,\mathrm{m}\) and \(b=3\,376\,200\,\mathrm{m}\) around its minor axis. What is the volume \(V\) of this ellipsoid?
\(V\doteq 1.631\cdot 10^{20}\,\mathrm{m}^3 \)
\(V\doteq 1.622\cdot 10^{20}\,\mathrm{m}^3 \)
\(V\doteq 3.602\cdot 10^{13}\,\mathrm{m}^3 \)
\(V\doteq 1.132\cdot 10^{14}\,\mathrm{m}^3 \)