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.
\(\frac{\pi}2(\sqrt{2}-1)\)
\(1.3\)
\(1.5\pi\)
\(0.3\)
Applications of definite integral
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.
\(2\pi(\sqrt{2}-1)\)
\(1.3\)
\(6\pi\)
\(5.2\)
2010012606
Level:
B
Part of the graph of the function \(f(x) = \frac{1}
{x^2}\)
is shown in the picture. Consider the region bounded by
\(x\)-axis,
graph of \(f\)
and
lines \(x = 1\) and
\(x = 2\). Find
the volume of the solid of revolution obtained by revolving this region about
\(x\)-axis.
\(\frac{7}
{24} \pi \)
\(\frac{\pi}
{2}\)
\(\frac{9}
{24} \pi \)
\(\frac{7}
{8} \pi \)
2010012605
Level:
B
The function \(f(x) = \frac12 x +2\)
is graphed in the picture. Consider the region between the graph of the function \(f\), the
\(x\)-axis and
the lines \(x = -2\)
and \(x = 1\).
Find the volume of the solid of revolution obtained by revolving this region about
\(x\)-axis.
\(\frac{39}
{4} \pi \)
\(\frac{55}
{4} \pi \)
\(3\pi \)
\(\frac{10}
{3} \pi \)
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 \(5\, \mathrm{m}\).
\(\frac{3}
{10}c\, \mathrm{J}\)
\(\frac{2}
{5}c\, \mathrm{J}\)
\(c\, \mathrm{J}\)
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?
\(108\, \mathrm{m}\)
\(54\, \mathrm{m}\)
\(324\, \mathrm{m}\)
2010012602
Level:
A
Find the area of the region bounded by the curves
\(y = 2^{x}\),
\(y = 2^{-x}\) and
\(y = 4\).
\(16 -\frac{6}
{\ln 2}\)
\(16 -\frac{10}
{\ln 2}\)
\(8 -\frac{5}
{\ln 2}\)
\(16 +\frac{6}
{\ln 2}\)
2010012601
Level:
A
Find the area of the region bounded by the curves
\(y =\mathrm{e} ^{x}-1\),
\(y = -\mathrm{e}^{x} + 1\) and
\(x = 1\).
\(2\mathrm{e}-4 \)
\(2\mathrm{e}-2\)
\(2\)
\(4-2\mathrm{e} \)
2010011407
Level:
A
Find the area of the region bounded by the curves
\(y = x+1\) and
\(y = -x^{2} + 3\).
\( \frac{9}2\)
\(2\)
\(6\)
\( \frac{21}2\)
2010011406
Level:
A
Find the area of the region bounded by the graph of
\(f(x)=\sin x-1\) on
\(\left [ \frac{\pi }{2};\pi \right ] \) and
lines \(y = 0\)
and \(x =\pi \).
\( \frac{\pi}2-1\)
\( \frac{\pi}2\)
\(1\)
\( \frac{3\pi}2-1\)