1003108904
Level:
A
What is the area of the plain region bounded by the lines $x=2$, $x=3$ and the graphs of functions $f(x)=\sin x$ and $g(x)=\frac1{x^2}$? Round the result to three decimal digits.
$0.407$
$0.167$
$1.573$
$0.623$
Applications of definite integral
1103108903
Level:
A
Which of the given pictures shows the shaded region of the biggest area?
1103108902
Level:
A
Which of the given expressions does not describe the area of the region highlighted in yellow? (See the picture.)
$\int\limits_1^3\frac1x\,\mathrm{d}x$
$2\int\limits_1^3\frac1x\,\mathrm{d}x$
$\int\limits_1^4\frac2x\,\mathrm{d}x-\int\limits_3^4\frac2x\,\mathrm{d}x$
$\int\limits_1^3\frac1x\,\mathrm{d}x-\int\limits_1^3-\frac1x\,\mathrm{d}x$
1103108901
Level:
A
Which of the given expressions does not describe the area of the region highlighted in yellow? (See the picture.)
$4\cdot\int\limits_0^{\frac{\pi}4}4\sin x\,\mathrm{d}x$
$4\cdot\int\limits_0^{\pi}\sin x\,\mathrm{d}x$
$8\cdot\int\limits_0^{\pi}\frac{\sin x}2\,\mathrm{d}x$
$\int\limits_0^{\frac{\pi}2}16\cdot\frac{\sin x}2\,\mathrm{d}x$
1003124410
Level:
A
Calculate the area of a plane region bounded by the graphs of the functions $f(x)=-\frac x{\pi}+1$ and $g(x)=-\sin x$, and by the lines $x=\pi$ and $x=0$.
$2+\frac{\pi}2$
$2-\frac{\pi}2$
$-2-\frac{\pi}2$
$\frac{2+\pi}2$
1003124409
Level:
C
Consider a conic section defined by $y^2=x+a$. Find the value of the real parameter $a$, if a solid of the volume $\frac{125\pi}2$ is obtained by rotating the given conic about the $x$-axis on the interval $[0;5]$.
$a=10$
$a=5$
$a=15$
$a=25$
1003124408
Level:
C
Consider a conic section defined by $y^2=x-5$. Find the value of the real parameter $a$, if a solid of the volume $\frac{9\pi}2$ is obtained by rotating the given conic about the $x$-axis on the interval $[5;a]$.
$a=8$
$a=2$
$a=7.5$
$a=9$
1003124407
Level:
C
Consider a conic section defined by $x^2-4y^2=1$. Find the volume of the solid obtained by rotating the given conic about the $x$-axis on the interval $[1;6]$.
$\frac{50}3\pi$
$\frac{35}4\pi$
$\frac{49}3\pi$
$\frac{50}3$
1003124406
Level:
A
Calculate the area of a plane region bounded by the graphs of the functions $f(x)=x^2+2x+2$ and $g(x)=6-x^2$.
$9$
$21$
$15$
$\frac{59}3$
1103124405
Level:
C
Let a solid be obtained by rotating the blue triangle about the $y$-axis (see the picture). Find the volume of this solid.
$32\pi$
$32$
$96\pi$
$100$