Applications of definite integral

1103108906

Level:

Find the area of the triangle

A B C

highlighted in yellow (see the picture).

20.5

22.5

20

21

1003108905

Level:

Find the area of the plain region bounded by the curves

f (x) = x^{5}

and

g (x) = x^{9}

on the interval

[1; 2]

91.8

113.2

91.6

100.4

1003108904

Level:

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) = \frac{1}{x^{2}}

? Round the result to three decimal digits.

0.407

0.167

1.573

0.623

1103108903

Level:

Which of the given pictures shows the shaded region of the biggest area?

1103108902

Level:

Which of the given expressions does not describe the area of the region highlighted in yellow? (See the picture.)

\int_{1}^{3} \frac{1}{x} d x

2 \int_{1}^{3} \frac{1}{x} d x

\int_{1}^{4} \frac{2}{x} d x - \int_{3}^{4} \frac{2}{x} d x

\int_{1}^{3} \frac{1}{x} d x - \int_{1}^{3} - \frac{1}{x} d x

1103108901

Level:

Which of the given expressions does not describe the area of the region highlighted in yellow? (See the picture.)

4 \cdot \int_{0}^{\frac{π}{4}} 4 \sin x d x

4 \cdot \int_{0}^{π} \sin x d x

8 \cdot \int_{0}^{π} \frac{\sin x}{2} d x

\int_{0}^{\frac{π}{2}} 16 \cdot \frac{\sin x}{2} d x

1003124410

Level:

Calculate the area of a plane region bounded by the graphs of the functions

f (x) = - \frac{x}{π} + 1

and

g (x) = - \sin x

, and by the lines

x = π

and

x = 0

2 + \frac{π}{2}

2 - \frac{π}{2}

- 2 - \frac{π}{2}

\frac{2 + π}{2}

1003124409

Level:

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 π}{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:

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 π}{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:

Consider a conic section defined by

x^{2} - 4 y^{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} π

\frac{35}{4} π

\frac{49}{3} π

\frac{50}{3}

Applications of definite integral

1103108906

1003108905

1003108904

1103108903

1103108902

1103108901

1003124410

1003124409

1003124408

1003124407

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