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Examples of solving problems to the section integral calculus

Lecture



Example N 1

Find the indefinite integral:

  Examples of solving problems to the section integral calculus
Decision.
  Examples of solving problems to the section integral calculus
Answer:
  Examples of solving problems to the section integral calculus

Example N 2

Find the indefinite integral:

  Examples of solving problems to the section integral calculus
Decision.
  Examples of solving problems to the section integral calculus
Answer:
  Examples of solving problems to the section integral calculus

Example N 3

Find the indefinite integral:

  Examples of solving problems to the section integral calculus .
Decision.

To integrate this function, it is necessary to make a change of variable   Examples of solving problems to the section integral calculus .

  Examples of solving problems to the section integral calculus

We got the wrong fraction. Select the integer part in it, dividing the polynomial standing in the numerator by the denominator polynomial by a corner.

  Examples of solving problems to the section integral calculus
Answer:
  Examples of solving problems to the section integral calculus

Example N 4

Find the indefinite integral:

  Examples of solving problems to the section integral calculus
Decision.
  Examples of solving problems to the section integral calculus

We got an irregular fraction; we single out the integer part in it, dividing the polynomial in the numerator into a denominator polynomial by a corner:

  Examples of solving problems to the section integral calculus
  Examples of solving problems to the section integral calculus
Answer:
  Examples of solving problems to the section integral calculus

Example N 5

Find the indefinite integral:

  Examples of solving problems to the section integral calculus
Decision.
  Examples of solving problems to the section integral calculus
Answer:
  Examples of solving problems to the section integral calculus

Example N 6

Find the indefinite integral:

  Examples of solving problems to the section integral calculus
Decision.
  Examples of solving problems to the section integral calculus
Answer:
  Examples of solving problems to the section integral calculus

Example N 7

Find the area of ​​the figure bounded by the lines:

  Examples of solving problems to the section integral calculus
Decision.
We construct a figure on the OXY plane bounded by y 1 = 4 - x 2 - by a parabola and y 2 = 2 - x - by a straight line (Fig. 16).

  Examples of solving problems to the section integral calculus

Fig. sixteen

Find the intersection points of the functions y 1 and y 2 :

  Examples of solving problems to the section integral calculus

Since 4 - x 2 ≤ 2 - x on the segment [-1; 2], the area S of this figure is calculated as follows:

  Examples of solving problems to the section integral calculus
The answer is 4.5 .

Example N 8

Find the length of the arc curve   Examples of solving problems to the section integral calculus .

Decision.
  Examples of solving problems to the section integral calculus
  Examples of solving problems to the section integral calculus
Answer:   Examples of solving problems to the section integral calculus

Example N 9

Find the length of the arc curve   Examples of solving problems to the section integral calculus .

Decision.

The arc is explicitly specified.   Examples of solving problems to the section integral calculus , the limits of integration are given 2 ≤ x ≤ 3, we make the integral

  Examples of solving problems to the section integral calculus
Answer:   Examples of solving problems to the section integral calculus

Example N 10

Find the length of the astroid arc

  Examples of solving problems to the section integral calculus .
Decision.

The curve is set parametrically, therefore, its length L is calculated as follows:

  Examples of solving problems to the section integral calculus
  Examples of solving problems to the section integral calculus
The answer is: 4.5

Example N 11

Find the volume of the body obtained by rotating around the OX axis of the figure, which is limited by parabolas y = 3 - x 2 , y = x 2 + 1.

Decision.

We construct a figure on the plane of the OXU , bounded by parabolas y = 3 - x 2 , y = x 2 + 1 (Fig. 17).

  Examples of solving problems to the section integral calculus

Fig. 17

Find the intersection points of the curves:

  Examples of solving problems to the section integral calculus .
Then
  Examples of solving problems to the section integral calculus
  Examples of solving problems to the section integral calculus
Answer:   Examples of solving problems to the section integral calculus

Example N 12

Change the order of integration:

  Examples of solving problems to the section integral calculus
Decision.

We build the integration domain D , which consists of two areas - D 1 and D 2 (Fig. 18):

  Examples of solving problems to the section integral calculus

Area D 1 is bounded by a semicircle   Examples of solving problems to the section integral calculus and straight lines y = 0, x = 2.

The domain D 2 is bounded by the straight lines y = x - 4, y = 0, x = 2.

  Examples of solving problems to the section integral calculus

Fig. 18

Region D is correct in the direction of the axis of the shelter . The largest value of y in the domain D is the number y = 0, and the smallest y = -2. From the equation   Examples of solving problems to the section integral calculus Express x through y :

  Examples of solving problems to the section integral calculus

Function   Examples of solving problems to the section integral calculus sets the left boundary of the domain D. From the equation y = x - 4 we express x through y : x = y + 4. The function x = y + 4 sets the right boundary of the domain D. Therefore, the domain D is given by the inequalities:   Examples of solving problems to the section integral calculus

Now we change the order of integration in the double integral:

  Examples of solving problems to the section integral calculus
Answer:
  Examples of solving problems to the section integral calculus

Example N 13

Find the area of ​​the figure bounded by lines   Examples of solving problems to the section integral calculus ,   Examples of solving problems to the section integral calculus ,   Examples of solving problems to the section integral calculus ,   Examples of solving problems to the section integral calculus .

Decision.

Let's build a shape D bounded by given lines (fig. 19):

  Examples of solving problems to the section integral calculus - circle centered at a point   Examples of solving problems to the section integral calculus radius   Examples of solving problems to the section integral calculus ;

  Examples of solving problems to the section integral calculus - a circle with a center at the point (1; 0), radius 1;

  Examples of solving problems to the section integral calculus - OX axis;

  Examples of solving problems to the section integral calculus - straight with the slope   Examples of solving problems to the section integral calculus

  Examples of solving problems to the section integral calculus

Fig. nineteen

Expressing circles in polar coordinates:

  Examples of solving problems to the section integral calculus

we conclude that the domain D can be given by the inequalities:   Examples of solving problems to the section integral calculus

Find the area S of the area D according to the formula of the area of ​​the figures in polar coordinates:

  Examples of solving problems to the section integral calculus
Answer:   Examples of solving problems to the section integral calculus

Example N 14

Calculate the volume of the body bounded by the surfaces   Examples of solving problems to the section integral calculus ,   Examples of solving problems to the section integral calculus ,   Examples of solving problems to the section integral calculus ,   Examples of solving problems to the section integral calculus .

Decision.

The body is bounded by parabolic cylinders, which form parallel to the axis OZ :   Examples of solving problems to the section integral calculus ,   Examples of solving problems to the section integral calculus ; coordinate plane   Examples of solving problems to the section integral calculus ; and the plane z = 4 - y . Let us depict the body (Fig. 20) and its projection (Fig. 21) in the DCS plane in the figures .

  Examples of solving problems to the section integral calculus   Examples of solving problems to the section integral calculus
Fig. 20 Fig. 21

The basis of a given body is the domain D , given by the inequalities:   Examples of solving problems to the section integral calculus .

Find the volume V of the body using a double integral:

  Examples of solving problems to the section integral calculus

First we find the inner integral:

  Examples of solving problems to the section integral calculus

Then we find the external integral:

  Examples of solving problems to the section integral calculus
Answer:   Examples of solving problems to the section integral calculus

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Mathematical analysis. Integral calculus

Terms: Mathematical analysis. Integral calculus