Applications of definite integral

9000100001

Level: 
B
The function \(f(x) = 3 - 2x\) is graphed in the picture. Consider the region between the graph of the function on the interval \([ 0;\, 1.5] \) and the axes. Determine the solid of revolution obtained by revolving this region about \(y\)-axis
A cone with the base of radius \(1.5\).
A cone with the base of radius \(3\).
A pyramid of the height \(1.5\).
A pyramid of the height \(3\).

9000100003

Level: 
B
The function \(f(x) = x^{2} + 2\) is graphed in the picture. Consider the region between the graph of the function on the interval \([ 0;\, 1] \), both axes and the line \(x = 1\). Find the formula for the volume of the solid of revolution obtained by revolving this region about \(y\)-axis.
\(V =\pi \int _{ 0}^{3}1\, \mathrm{d}y -\pi \int _{2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)
\(V =\pi \int _{ 0}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)
\(V =\pi \int _{ 2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y -\pi \int _{0}^{3}1\, \mathrm{d}y\)
\(V =\pi \int _{ 2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)

9000100007

Level: 
B
The function \(f(x)= \sqrt{x}\) is graphed in the picture. Consider the region bounded by the graph of \(f\) on \([ 1;\, 4] \), lines \(x = 1\), \(x = 4\) and the \(x\)-axis. Find the volume of the solid of revolution obtained by revolving this region about the \(x\)-axis.
\(\frac{15} {2} \pi \)
\(\frac{17} {2} \pi \)
\(\frac{17} {2} \pi ^{2}\)
\(\frac{15} {2} \pi ^{2}\)

9000100006

Level: 
B
The function \(f(x)= \sqrt{x}\) is graphed in the picture. Consider the region bounded by the graph of \(f\) on \([ 1;\, 4] \), lines \(x = 1\), \(x = 4\) and the \(x\)-axis. Identify the formula for volume of the solid of revolution obtained by revolving this region about the \(x\)-axis.
\(V =\pi \int _{ 1}^{4}x\, \mathrm{d}x\)
\(V =\int _{ 1}^{4}x\, \mathrm{d}x\)
\(V =\pi \int _{ 1}^{4}\sqrt{x}\, \mathrm{d}x\)
\(V =\int _{ 1}^{4}\sqrt{x}\, \mathrm{d}x\)

9000100004

Level: 
B
The function \(f(x)= x^{2} + 2\) is graphed in the picture. Consider the region bounded by the graph of the function, both axes and the line \(x = -1\). Determine the solid of revolution obtained by revolving this region about \(x\)-axis.
A general solid which is neither cone nor cylinder.
Cone with base of radius \(1\).
Cylinder with base of radius \(2\).
Cone with base of radius \(2\).

9000100005

Level: 
B
The function \(f(x) = 1\) is graphed in the picture. Determine the solid of revolution with volume given by the following formula. \[ \pi \int _{-1}^{1}f^{2}(x)\, \mathrm{d}x \]
Cylinder of base radius \(1\) and height \(2\).
Cone of base radius \(1\) and height \(2\).
Cone of base radius \(2\) and height \(1\).
Cylinder of base radius \(2\) and height \(1\).

9000100008

Level: 
B
Part of the graph of the function \(f(x) = \frac{1} {x}\) is shown in the picture. Complete the following sentence: „Formula \[ V =\pi \int _{ 1}^{2}x^{-2}\, \mathrm{d}x \] determines the volume of the solid of revolution obtained by revolving region bounded by ...”
\(x\)-axis, graph of \(f\) on \([ 1;\, 2] \) and lines \(x = 1\), \(x = 2\) about \(x\)-axis.
\(y\)-axis, graph of \(f\) on \([ 1;\, 2] \) and lines \(y = 1\), \(y = \frac{1} {2}\) about \(x\)-axis.
\(x\)-axis, graph of \(f^{2}\) on \([ 1;\, 2] \) and lines \(x = 1\), \(x = 2\) about \(x\)-axis.
\(y\)-axis, graph of \(f^{2}\) on \([ 1;\, 2] \) and lines \(y = 1\), \(y = \frac{1} {2}\) about \(x\)-axis.

9000100002

Level: 
B
The function \(f(x) = 3 - 2x\) is graphed in the picture. Consider the region between the graph of the function \(f\), the \(x\)-axis and the lines \(x = 1\) and \(x = -1\). Find the volume of the solid of revolution obtained by revolving this region about \(x\)-axis.
\(\frac{62} {3} \pi \)
\(6\pi \)
\(12\pi \)
\(\frac{8} {3}\pi \)

9000100009

Level: 
B
Part of the graph of the function \(f(x) = \frac{1} {x}\) is shown in the picture. Consider the region bounded by \(x\)-axis, graph of \(f\) and lines \(x = 1\) and \(x = 4\). Find the volume of the solid of revolution obtained by revolving this region about \(x\)-axis.
\(\frac{3} {4}\pi \)
\(\frac{5} {4}\pi \)
\(\frac{5} {3}\pi \)
\(\frac{4} {3}\pi \)

9000072908

Level: 
C
A \(100\, \mathrm{kg}\) heavy anchor is attached to a \(20\, \mathrm{m}\) long rope. One meter of the rope weights \(1\, \mathrm{kg}\). Find the work required to raise the anchor \(20\, \mathrm{m}\) higher. The standard acceleration of gravity is \(9.81\, \mathrm{m\, s}^{-2}\). Neglect the buoyancy (the force from the Archimedes law).
\(21\: 582\, \mathrm{J}\)
\(23\: 544\, \mathrm{J}\)
\(19\: 620\, \mathrm{J}\)