Applications of definite integral

1103068001

Level: 
A
Which of the following formulas does NOT express the area of the yellow triangle indicated in the picture?
\( \int\limits_1^ 6(-0.8x+5.8)\,\mathrm{d}x \)
\( \frac12\cdot(5-1)\cdot(6-1)\cdot\sin90^{\circ} \)
\( \frac{4\cdot5}2 \)
\( \int\limits_1^ 6(-0.8x+5.8)\,\mathrm{d}x-5 \)

1003118706

Level: 
B
Consider a truncated cone with the diameters of its bases \( 2\,\mathrm{cm} \) and \( 10\,\mathrm{cm} \), and with the height \( 4\,\mathrm{cm} \). Which of the following formulas cannot be used to calculate the volume of such truncated cone?
\( V=\pi\int\limits_0^4(5-x)\,\mathrm{d}x \)
\( V=\pi\int\limits_0^4(5-x)^2\mathrm{d}x \)
\( V=\frac{\pi}3\cdot4\cdot(25+5+1) \)
\( V=\frac{\pi}3\cdot25\cdot5-\frac{\pi}3\cdot1\cdot1 \)

1003118705

Level: 
B
Peter and Jane both calculated the volume of a solid of revolution using a definite integral. Both chose a solid obtained by rotation of a line segment about the \( x \)-axis. The endpoints of Peter’s line segment are \( [0;1] \) and \( [5;4] \), the endpoints of Jane’s line segment are \( [0;3] \) and \( [5;0] \). Finally, they compared their calculated volumes. Which of the following statements is true?
Peter’s solid is \( 20\pi \) bigger.
Jane’s solid is \( 20\pi \) bigger.
Both solids have the same volume.
The difference between Peter’s solid and Jane’s solid is \( 10\pi \).

1103118704

Level: 
B
Which of the given equations defines the line that together with \( x=0 \) and \( x \)-axis bounds the right triangle, if by rotation of this triangle about the \( x \)-axis the cone of height \( 10 \) is obtained as indicated in the picture?
\( y=-0.4x+4 \)
\( y=-2.5x+10 \)
\( y=4x+10 \)
\( y=10x+4 \)

1003118703

Level: 
B
A right trapezoid is bounded by \( y=ax+1 \), \( x=0 \), \( x=6 \), and by the \( x \)-axis. Rotating the trapezoid around the \( x \)-axis we get a truncated cone. Find the value of the parameter \( a > 0 \) so that the volume of the truncated cone is \( 26\pi \).
\( a=\frac13 \)
\( a=\frac12 \)
\( a=3 \)
\( a=2 \)

1003118702

Level: 
B
It is possible to calculate the volume of a sphere with the radius of \( 3 \) using a definite integral. Which of the following formulas is not correct?
\( \int\limits_{-3}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( \pi\int\limits_{-3}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( 2\pi\int\limits_{0}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( \pi\int\limits_{-3}^3\left(-\sqrt{9-x^2}\right)^2\,\mathrm{d}x \)

1103068303

Level: 
B
Which of the following formulas will not give the volume of the solid created by the rotation of the red area about the \( x \)-axis (see the picture)?
\( \pi\cdot\int\limits_{\frac{\pi}4}^{\frac{3\pi}4}\sin x\,\mathrm{d}x \)
\( \pi\cdot\int\limits_{\frac{\pi}4}^{\frac{3\pi}4}\sin^2⁡x\,\mathrm{d}x \)
\( 2\pi\cdot\int\limits_{\frac{\pi}2}^{\frac{3\pi}4}\sin^2x\,\mathrm{d}x \)
\( \pi\cdot\int\limits_{\frac{9\pi}4}^{\frac{11\pi}4}\sin^2x\,\mathrm{d}x \)