Applications of definite integral

2010014706

Level: 
C
In the experimental process, the ideal gas is expanded adiabatically from an initial volume of \(V_1=0.3\,\mathrm{m}^3\) to a final volume of \(V_2=0.8\,\mathrm{m}^3\). Find a work done by the gas in the given process. Hint: An adiabatic process with an ideal gas follows the relationship \(pV^{1.4}=c\), where \(p\) is a gas pressure, \(V\) is a gas volume, and \(c\) is a positive constant. The work \(W\) done by a gas is defined as \(W=\int_{V_1}^{V_2}p\mathrm{d}V\).
\( W\doteq 1.313c\,\mathrm{J}\)
\( W \doteq 0.375c\,\mathrm{J}\)
\( W \doteq 6.782c\,\mathrm{J}\)
\( W \doteq 0.221c\,\mathrm{J}\)

2010014705

Level: 
C
In the experimental process, the ideal gas is expanded isothermally from an initial pressure of \(0.8\,\mathrm{MPa}\) and volume of \(V_1=0.3\,\mathrm{m}^3\) to a final volume of \(V_2=1.2\,\mathrm{m}^3\). Find a work done by the gas in the given process. Hint: During isothermal expansion, both pressure \(p\) and volume \(V\) change along an isotherm with a constant \(pV\) product. The work \(W\) done by a gas is defined as \(W=\int_{V_1}^{V_2}p\mathrm{d}V\).
\( W\doteq 333\,\mathrm{kJ}\)
\( W \doteq 216\,\mathrm{kJ}\)
\( W \doteq 720\,\mathrm{kJ}\)
\( W \doteq 178\,\mathrm{kJ}\)

2010014704

Level: 
C
The time characteristic of an alternating current \(i\) is given in the figure. Find the effective value \(I\) of the alternating current \(i\) provided the following relation holds: \(I^2T=\int_0^T i^2\mathrm{d}t\).
\( I=500\,\mathrm{mA}\)
\( I=354\,\mathrm{mA}\)
\( I=0\,\mathrm{mA}\)
\( I=250\,\mathrm{mA}\)

2010014703

Level: 
C
The time characteristic of an alternating voltage \(u\) is given in the figure. Find the effective value \(U\) of the alternating voltage \(u\) provided the following relation holds: \(U^2T=\int_0^T u^2\mathrm{d}t\).
\( U=325\,\mathrm{V}\)
\( U\doteq 230\,\mathrm{V}\)
\( U=0\,\mathrm{V}\)
\( U=\frac{325}2\,\mathrm{V}\)

2010014702

Level: 
C
The time characteristic of an alternating voltage \(u\) is given in the figure, where \(U_m\) is the peak value of \(u\). Find the effective value \(U\) of the alternating voltage \(u\) provided the following relation holds: \(U^2T=\int_0^T u^2\mathrm{d}t\).
\( U=\frac{\sqrt{3}}3 U_m\)
\( U=\frac{\sqrt{2}}2 U_m\)
\( U=\frac{1}3 U_m\)
\( U=\frac{1}2 U_m\)

2010014701

Level: 
C
The time characteristic of an alternating current \(i\) is given in the figure, where \(I_m\) is the peak value of \(i\). Find the effective value \(I\) of the alternating current \(i\) provided the following relation holds: \(I^2T=\int_0^T i^2\mathrm{d}t\).
\( I=\frac{\sqrt{3}}3 I_m\)
\( I=\frac{\sqrt{2}}2 I_m\)
\( I=\frac{1}3 I_m\)
\( I=\frac{1}2 I_m\)

2010014306

Level: 
C
Approximately, the shape of Mars is an ellipsoid. This ellipsoid can be obtained by rotating an ellipse with semi-axes \(a=3\,396\,190\,\mathrm{m}\) and \(b=3\,376\,200\,\mathrm{m}\) around its minor axis. What is the volume \(V\) of this ellipsoid?
\(V\doteq 1.631\cdot 10^{20}\,\mathrm{m}^3 \)
\(V\doteq 1.622\cdot 10^{20}\,\mathrm{m}^3 \)
\(V\doteq 3.602\cdot 10^{13}\,\mathrm{m}^3 \)
\(V\doteq 1.132\cdot 10^{14}\,\mathrm{m}^3 \)

2010014305

Level: 
C
Approximately, the shape of the Earth is an ellipsoid. This ellipsoid can be obtained by rotating an ellipse with semi-axes \(a=6\,378\,137\,\mathrm{m}\) and \(b=6\,356\,752\,\mathrm{m}\) around its minor axis. What is the volume \(V\) of this ellipsoid?
\(V\doteq 1.083\cdot 10^{21}\,\mathrm{m}^3 \)
\(V\doteq 1.080\cdot 10^{21}\,\mathrm{m}^3 \)
\(V\doteq 4.002\cdot 10^{14}\,\mathrm{m}^3 \)
\(V\doteq 1.274\cdot 10^{14}\,\mathrm{m}^3 \)