B

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 \(1\).

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 \(3\).

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.

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.
\(V =\pi \int _{ 0}^{3}1\, \mathrm{d}y -\pi \int _{2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)
\(V =\pi \int _{ 0}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)
\(V =\pi \int _{ 2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y -\pi \int _{0}^{3}1\, \mathrm{d}y\)
\(V =\pi \int _{ 2}^{3}(\sqrt{y - 2})^{2}\, \mathrm{d}y\)

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.
\(\frac{3} {4}\pi \)
\(\frac{5} {4}\pi \)
\(\frac{5} {3}\pi \)
\(\frac{4} {3}\pi \)

9000086607

Level: 
B
Determine the truth values of propositions \(a\) and \(b\) if you know that the compound proposition \[ (\neg a \vee b) \wedge a \] is true.
Both statements are true.
The statement \(a\) is true, \(b\) is false.
The statement \(a\) is false, \(b\) is true.
Both statements are false.