Applications of definite integral

9000065610

Level: 
A
Using definite integral find the area of the triangle defined by the following three inequalities \[ \begin{aligned}y& > 0, & \\y& < x + 3, \\y& < 3 - x. \\ \end{aligned} \]
\(\int _{-3}^{0}(x + 3)\, \mathrm{d}x +\int _{ 0}^{3}(3 - x)\, \mathrm{d}x\)
\(\int _{0}^{3}(x + 3)\, \mathrm{d}x\)
\(\int _{-3}^{3}(3 - x)\, \mathrm{d}x\)
\(\int _{-3}^{0}(3 - x)\, \mathrm{d}x +\int _{ 0}^{3}(x + 3)\, \mathrm{d}x\)

1003068201

Level: 
B
The value of the integral \[ \frac{4\pi}9\int\limits_0^3 x^2\mathrm{d}x \] is a number which represents:
the volume of a cone with the base radius of \( 2\,\mathrm{cm} \) and the height of \( 3\,\mathrm{cm} \).
the volume of a cone with the base radius of \( 3\,\mathrm{cm} \) and the height of \( 2\,\mathrm{cm} \).
the volume of a sphere segment which is a part of a sphere with the radius of \( \frac23\,\mathrm{cm} \) and the height of \( 3\,\mathrm{cm} \).
volume of a sphere segment which is a part of a sphere with the radius of \( 3\,\mathrm{cm} \) and the height of \( \frac23\,\mathrm{cm} \).

1003068202

Level: 
B
The value of the integral \[ \pi\cdot\int\limits_0^6\left[9-(x-3)^2\right]\,\mathrm{d}x \] is a number which represents:
the volume of a sphere with the radius of \( 3\,\mathrm{cm} \).
the volume of a sphere with the radius of \( 6\,\mathrm{cm} \).
the volume of a sphere with the diameter of \( 3\,\mathrm{cm} \).
the volume of a semi-sphere with the radius of \( 3\,\mathrm{cm} \).

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?
\( \int\limits_{-3}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( \pi\int\limits_{-3}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( 2\pi\int\limits_{0}^3\left(9-x^2\right)\,\mathrm{d}x \)
\( \pi\int\limits_{-3}^3\left(-\sqrt{9-x^2}\right)^2\,\mathrm{d}x \)

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 \( 26\pi \).
\( a=\frac13 \)
\( a=\frac12 \)
\( a=3 \)
\( a=2 \)

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 \( 10\pi \).

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?
\( V=\pi\int\limits_0^4(5-x)\,\mathrm{d}x \)
\( V=\pi\int\limits_0^4(5-x)^2\mathrm{d}x \)
\( V=\frac{\pi}3\cdot4\cdot(25+5+1) \)
\( V=\frac{\pi}3\cdot25\cdot5-\frac{\pi}3\cdot1\cdot1 \)

1103068301

Level: 
B
Which of these formulas can be used to find the volume of the cone in the picture?
\( \pi\cdot\int\limits_0^4\left(-\frac14x+1\right)^2\,\mathrm{d}x \)
\( \pi\cdot\int\limits_0^1\left(-4x+4\right)^2\,\mathrm{d}x \)
\( 2\pi\cdot\int\limits_0^4\left(-\frac14x+1\right)^2\,\mathrm{d}x \)
\( 2\pi\cdot\int\limits_0^1\left(-4x+4\right)^2\,\mathrm{d}x \)

1103068302

Level: 
B
Which of these formulas can be used to find the volume of the cylinder in the picture? Points \( [0; 0; 0] \) and \( [4;0;0] \) are located in the centers of the cylinder bases.
\( \pi\cdot\int\limits_0^43^2\,\mathrm{d}x \)
\( \pi\cdot\int\limits_0^34^2\,\mathrm{d}x \)
\( \pi\cdot\int\limits_0^43\,\mathrm{d}x \)
\( \pi\cdot\int\limits_{-4}^49\,\mathrm{d}x \)