Applications of definite integral

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: \(I^2T=\int_0^T i^2\mathrm{d}t\).
\( I=\frac{\sqrt{3}}3 I_m\)
\( I=\frac{\sqrt{2}}2 I_m\)
\( I=\frac{1}3 I_m\)
\( I=\frac{1}2 I_m\)

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: \(U^2T=\int_0^T u^2\mathrm{d}t\).
\( U=\frac{\sqrt{3}}3 U_m\)
\( U=\frac{\sqrt{2}}2 U_m\)
\( U=\frac{1}3 U_m\)
\( U=\frac{1}2 U_m\)

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: \(U^2T=\int_0^T u^2\mathrm{d}t\).
\( U=325\,\mathrm{V}\)
\( U\doteq 230\,\mathrm{V}\)
\( U=0\,\mathrm{V}\)
\( U=\frac{325}2\,\mathrm{V}\)

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: \(I^2T=\int_0^T i^2\mathrm{d}t\).
\( I=500\,\mathrm{mA}\)
\( I=354\,\mathrm{mA}\)
\( I=0\,\mathrm{mA}\)
\( I=250\,\mathrm{mA}\)

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 \(W=\int_{V_1}^{V_2}p\mathrm{d}V\).
\( W\doteq 333\,\mathrm{kJ}\)
\( W \doteq 216\,\mathrm{kJ}\)
\( W \doteq 720\,\mathrm{kJ}\)
\( W \doteq 178\,\mathrm{kJ}\)

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 \(W=\int_{V_1}^{V_2}p\mathrm{d}V\).
\( W\doteq 1.313c\,\mathrm{J}\)
\( W \doteq 0.375c\,\mathrm{J}\)
\( W \doteq 6.782c\,\mathrm{J}\)
\( W \doteq 0.221c\,\mathrm{J}\)

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 \(t = 9\, \mathrm{s}\).
\(132\, \mathrm{m}\)
\(4\left (4 + \sqrt{2}\right )\mathrm{m}\)
\(10\, \mathrm{m}\)

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?
\(32\, \mathrm{m}\)
\(48\, \mathrm{m}\)
\(24\, \mathrm{m}\)

9000072903

Level: 
C
The force required to deform a spring is proportional to the extension of the spring. The current elongation of the spring is \(2\, \mathrm{cm}\) and the force required to reach this elongation is \(3\, \mathrm{N}\). Evaluate the work required to stretch the spring from the current elongation (i.e. \(2\, \mathrm{cm}\)) by additional \(10\, \mathrm{cm}\).
\(1.05\, \mathrm{J}\)
\(0.75\, \mathrm{J}\)
\(0.18\, \mathrm{J}\)

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 \(1\, \mathrm{m}\).
\(\frac{2} {3}c\, \mathrm{J}\)
\(\frac{1} {3}c\, \mathrm{J}\)
\(c\, \mathrm{J}\)