Applications of definite integral

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

9000072902

Level: 
C
The instantaneous velocity of a moving body is proportional to the square of the time. The velocity at the time \(t = 2\, \mathrm{s}\) is \(v = 6\, \mathrm{m\, s}^{-1}\). What is the distance traveled by the body in the first \(4\) seconds?
\(32\, \mathrm{m}\)
\(48\, \mathrm{m}\)
\(24\, \mathrm{m}\)

9000072904

Level: 
C
The force of the repulsion of two charged 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 \(3\, \mathrm{m}\) to \(1\, \mathrm{m}\).
\(\frac{2} {3}c\, \mathrm{J}\)
\(\frac{1} {3}c\, \mathrm{J}\)
\(c\, \mathrm{J}\)

9000072901

Level: 
C
The velocity of a moving body in meters per second is given by the function \(v(t) = 3\sqrt{t} + 2t\), where \(t\) is a time measured in seconds. Find the distance traveled by the body in the time interval from \(t = 1\, \mathrm{s}\) to \(t = 9\, \mathrm{s}\).
\(132\, \mathrm{m}\)
\(4\left (4 + \sqrt{2}\right )\mathrm{m}\)
\(10\, \mathrm{m}\)