1003124406
Level:
A
Calculate the area of a plane region bounded by the graphs of the functions $f(x)=x^2+2x+2$ and $g(x)=6-x^2$.
$9$
$21$
$15$
$\frac{59}3$
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.
$32\pi$
$32$
$96\pi$
$100$
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 $48\pi$.
$a=4$
$a=2$
$a=3$
$a=6$
1103124403
Level:
B
Let a solid be obtained by rotating the yellow region about the $x$-axis (see the picture). Find its volume.
$85.5\pi$
$85.5$
$27$
$27\pi$
1103124402
Level:
C
Find such a real number $a$, that the area of the region highlighted in green equals $6$ (see the picture).
$a=2$
$a=1$
$a=3$
$a=4$
1103124401
Level:
A
Find the area of the region highlighted in red (see the picture).
$1.5\pi$
$2\pi$
$2.5\pi$
$4.5\pi$
1003068203
Level:
B
What is the volume of the solid that we get by rotating the curve
\[ y=\frac1{x^2} \]
about the \( x \)-axis on the interval \( [1;3] \)?
\( \frac{26}{81}\pi \)
\( \frac{3^5-1}{3^6}\pi \)
\( \frac{28}{81}\pi \)
\( \frac{3^5+1}{3^6}\pi \)
1003068202
Level:
B
The value of the integral
\[ \pi\cdot\int\limits_0^6\left[9-(x-3)^2\right]\,\mathrm{d}x \]
is a number which represents:
the volume of a sphere with the radius of \( 3\,\mathrm{cm} \).
the volume of a sphere with the radius of \( 6\,\mathrm{cm} \).
the volume of a sphere with the diameter of \( 3\,\mathrm{cm} \).
the volume of a semi-sphere with the radius of \( 3\,\mathrm{cm} \).
1003068201
Level:
B
The value of the integral
\[ \frac{4\pi}9\int\limits_0^3 x^2\mathrm{d}x \]
is a number which represents:
the volume of a cone with the base radius of \( 2\,\mathrm{cm} \) and the height of \( 3\,\mathrm{cm} \).
the volume of a cone with the base radius of \( 3\,\mathrm{cm} \) and the height of \( 2\,\mathrm{cm} \).
the volume of a sphere segment which is a part of a sphere with the radius of \( \frac23\,\mathrm{cm} \) and the height of \( 3\,\mathrm{cm} \).
volume of a sphere segment which is a part of a sphere with the radius of \( 3\,\mathrm{cm} \) and the height of \( \frac23\,\mathrm{cm} \).
1103068005
Level:
A
Find the missing real constant \( a \) so that the green area and the red area indicated in the picture do equal.
\( a=-2\pi \)
\( a=-\frac32\pi \)
\( a=-\frac{\pi}2 \)
\( a=-3\pi \)