1103068304
Level:
B
Find the volume of the solid obtained by the rotation of the yellow area about the \( x \)-axis (see the picture).
\( \frac56\pi \)
\( \frac76\pi \)
\( \pi\cdot\ln 6 \)
\( \frac{35}{36}\pi \)
Applications of definite integral
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)?
\( \pi\cdot\int\limits_{\frac{\pi}4}^{\frac{3\pi}4}\sin x\,\mathrm{d}x \)
\( \pi\cdot\int\limits_{\frac{\pi}4}^{\frac{3\pi}4}\sin^2x\,\mathrm{d}x \)
\( 2\pi\cdot\int\limits_{\frac{\pi}2}^{\frac{3\pi}4}\sin^2x\,\mathrm{d}x \)
\( \pi\cdot\int\limits_{\frac{9\pi}4}^{\frac{11\pi}4}\sin^2x\,\mathrm{d}x \)
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.
\( \pi\cdot\int\limits_0^43^2\,\mathrm{d}x \)
\( \pi\cdot\int\limits_0^34^2\,\mathrm{d}x \)
\( \pi\cdot\int\limits_0^43\,\mathrm{d}x \)
\( \pi\cdot\int\limits_{-4}^49\,\mathrm{d}x \)
1103068301
Level:
B
Which of these formulas can be used to find the volume of the cone in the picture?
\( \pi\cdot\int\limits_0^4\left(-\frac14x+1\right)^2\,\mathrm{d}x \)
\( \pi\cdot\int\limits_0^1\left(-4x+4\right)^2\,\mathrm{d}x \)
\( 2\pi\cdot\int\limits_0^4\left(-\frac14x+1\right)^2\,\mathrm{d}x \)
\( 2\pi\cdot\int\limits_0^1\left(-4x+4\right)^2\,\mathrm{d}x \)
1103067903
Level:
A
Find the area of the yellow region indicated in the picture.
\( 8\pi \)
\( 6\pi \)
\( 2\pi \)
\( 10\pi \)
1103067902
Level:
A
Find the area of the region bounded by the curves \( y=x^3-5x^2+3x+5\text{, }x=0\text{, }x=1\text{, }y=-x+5 \) as indicated in the picture.
\( \frac7{12} \)
\( \frac{41}{12} \)
\( \frac{47}{12} \)
\( \frac5{12} \)
1103067901
Level:
A
Find the area of the yellow region indicated in the picture.
\( 6 \)
\( 0 \)
\( 3 \)
\( -6 \)
1103068103
Level:
A
Find the area of the yellow region indicated in the picture.
\( 2\sqrt2 \)
\( 2\sqrt3 \)
\( \sqrt2 \)
\( 4\sqrt2 \)
1103068102
Level:
A
Find the area of the yellow triangle ABC indicated in the picture. Read all the needed values from the picture.
\( 10 \)
\( 11 \)
\( 9.5 \)
\( 10.5 \)
1103068101
Level:
A
Find the area of the yellow region that is bounded by the parabola and the line as is indicated in the picture. Read all the needed values from the picture.
\( 4.5 \)
\( 22.5 \)
\( 3.75 \)
\( 10.25 \)