Definition 3. A body of revolution is a body obtained by rotating a flat figure around an axis that does not intersect the figure and lies in the same plane with it.

The axis of rotation can also intersect the figure if it is the axis of symmetry of the figure.

Theorem 2.
, axis
and straight line segments
and

rotates around an axis
. Then the volume of the resulting body of revolution can be calculated by the formula

(2)

Proof. For such a body, the section with the abscissa is a circle of radius
, means
and formula (1) gives the desired result.

If the figure is limited by the graphs of two continuous functions
and
, and line segments
and
, moreover
and
, then when rotating around the abscissa axis, we get a body whose volume

Example 3 Calculate the volume of a torus obtained by rotating a circle bounded by a circle

around the x-axis.

R solution. The specified circle is bounded from below by the graph of the function
, and above -
. The difference of the squares of these functions:

Desired volume

(the graph of the integrand is the upper semicircle, so the integral written above is the area of ​​the semicircle).

Example 4 Parabolic segment with base
, and height , revolves around the base. Calculate the volume of the resulting body ("lemon" by Cavalieri).

R solution. Place the parabola as shown in the figure. Then its equation
, and
. Let's find the value of the parameter :
. So, the desired volume:

Theorem 3. Let a curvilinear trapezoid bounded by the graph of a continuous non-negative function
, axis
and straight line segments
and
, moreover
, rotates around an axis
. Then the volume of the resulting body of revolution can be found by the formula

(3)

proof idea. Splitting the segment
dots

, into parts and draw straight lines
. The whole trapezoid will decompose into strips, which can be considered approximately rectangles with a base
and height
.

The cylinder resulting from the rotation of such a rectangle is cut along the generatrix and unfolded. We get an “almost” parallelepiped with dimensions:
,
and
. Its volume
. So, for the volume of a body of revolution we will have an approximate equality

To obtain exact equality, we must pass to the limit at
. The sum written above is the integral sum for the function
, therefore, in the limit we obtain the integral from formula (3). The theorem has been proven.

Remark 1. In Theorems 2 and 3, the condition
can be omitted: formula (2) is generally insensitive to the sign
, and in formula (3) it suffices
replaced by
.

Example 5 Parabolic segment (base
, height ) revolves around the height. Find the volume of the resulting body.

Decision. Arrange the parabola as shown in the figure. And although the axis of rotation crosses the figure, it - the axis - is the axis of symmetry. Therefore, only the right half of the segment should be considered. Parabola equation
, and
, means
. We have for volume:

Remark 2. If the curvilinear boundary of a curvilinear trapezoid is given by the parametric equations
,
,
and
,
then formulas (2) and (3) can be used with the replacement on the
and
on the
when it changes t from
before .

Example 6 The figure is bounded by the first arc of the cycloid
,
,
, and the abscissa axis. Find the volume of the body obtained by rotating this figure around: 1) the axis
; 2) axles
.

Decision. 1) General formula
In our case:

2) General formula
For our figure:

We encourage students to do all the calculations themselves.

Remark 3. Let a curvilinear sector bounded by a continuous line
and rays
,

, rotates around the polar axis. The volume of the resulting body can be calculated by the formula.

Example 7 Part of a figure bounded by a cardioid
, lying outside the circle
, rotates around the polar axis. Find the volume of the resulting body.

Decision. Both lines, and hence the figure they limit, are symmetrical about the polar axis. Therefore, it is necessary to consider only the part for which
. The curves intersect at
and

at
. Further, the figure can be considered as the difference of two sectors, and hence the volume can be calculated as the difference of two integrals. We have:

Tasks for an independent solution.

1. A circular segment whose base
, height , revolves around the base. Find the volume of the body of revolution.

2. Find the volume of a paraboloid of revolution whose base , and the height is .

3. Figure bounded by an astroid
,
rotates around the x-axis. Find the volume of the body, which is obtained in this case.

4. Figure bounded by lines
and
rotates around the x-axis. Find the volume of the body of revolution.

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bodies of revolution call bodies bounded either by a surface of revolution, or by a surface of revolution and a plane (Figure 134). Under the surface of revolution is understood the surface obtained from the rotation of a line ( ABCDE ), flat or spatial, called generatrix, around a fixed line ( i ) - axes of rotation.

Figure 134

Any point on the generatrix of the surface of rotation describes a circle located in a plane perpendicular to the axis of rotation - parallel, therefore, the plane perpendicular to the axis of revolution always intersects the surface of revolution in a circle. The greatest parallel - equator. The smallest parallel - throat(neck).

The planes passing through the axis of rotation are called meridional planes.

In a complex drawing, the representation of bodies of revolution is carried out by means of the representation of the edges of the bases and lines of the surface outlines.

The lines of intersection of meridional planes with the surface are called meridians.

The meridional plane parallel to the projection plane is called main meridional plane. The line of its intersection with the surface - prime meridian.

Straight circular cylinder. A right circular cylinder (Figure 135) is a body bounded by a cylindrical surface of revolution and two circles - the bases of the cylinder located in planes perpendicular to the axis of the cylinder. Cylindrical surface of revolution called the surface obtained by rotating a rectilinear generatrix AA 1 around a fixed straight line parallel to it - i (axis of rotation). The dimensions characterizing a straight circular cylinder are its diameter Dc and height l (distance between the bases of the cylinder).

Figure 135

A right circular cylinder can also be considered as a body obtained by rotating a rectangle. ABCD around one of its sides, for example, sun (Figure 136). Side sun is the axis of rotation, and the side AD - generatrix of the cylinder. The other two sides will mark the bases of the cylinder.

Figure 136

Rectangle AB and CD when rotated, they form circles - the bases of the cylinder.

Construction of cylinder projections.

The construction of horizontal and frontal projections of the cylinder begins with the image of the base of the cylinder, i.e., two projections of the circle (see Figure 135, b). Since the circle is in a plane H , then it is projected onto this plane without distortion. The frontal projection of a circle is a segment of a horizontal straight line equal to the diameter of the base circle.

After building the base on the frontal projection, two sketch generators(extreme generators) and the height of the cylinder is plotted on them. A segment of a horizontal line is drawn, which is a frontal projection of the upper base of the cylinder (Figure 135, c).

Determination of the missing projections of points A and B located on the surface of the cylinder, according to given frontal projections in this case does not cause difficulties, since the entire horizontal projection of the lateral surface of the cylinder is a circle (Figure 137, a). Therefore, the horizontal projections of the points BUT and AT can be found by swiping from the given points A"" and B"" vertical communication lines until they intersect with the circle at the desired points A" and B".

Profile projections of points BUT and AT They are also built using vertical and horizontal communication lines.

Isometric view of a cylinder draw, as shown in Figure 137, b.

In isometric point BUT and AT built according to their coordinates. For example, to build a point AT from the origin O along the axis x postpone the coordinate ∆x , and then a straight line is drawn through its end, parallel to the axis at , until it intersects with the base contour at the point 2 . From this point, a straight line is drawn parallel to the z axis, on which the coordinate is plotted Z B , points AT .

Figure 137

Straight circular cone . A right circular cone (Figure 138) is a body bounded by a conical surface of revolution and a circle located in a plane perpendicular to the axis of the cone. conical surface obtained by rotating a rectilinear generatrix SA (Figure 138, a), passing through fixed pointS on the axis of rotation i and making some constant angle with this axis. Dot S called the top of the cone, and the conical surface is the lateral surface of the cone. The size of a right circular cone characterizes the diameter of its base D K and height H .

Figure 138

A right circular cone can also be considered as a body obtained by rotating a right triangle SAB around his leg SB (Figure 139). With this rotation, the hypotenuse describes conical surface, and the leg AB - circle, i.e. the base of the cone.

Figure 139

Construction of cone projections.

The sequence of constructing two projections of the cone is shown in Figure 167, b and c. First, two projections of the base are built. The horizontal projection of the base is a circle. The frontal projection will be a segment of a horizontal line equal to the diameter of this circle (Figure 138, b). On the frontal projection, a perpendicular is erected from the middle of the base, and the height of the cone is laid on it (Figure 138, c). The resulting frontal projection of the top of the cone is connected by straight lines with the ends of the frontal projection of the base and a frontal projection of the cone is obtained.

Constructing points on the surface of a cone

If one point projection is given on the surface of the cone BUT (for example, the frontal projection in Figure 140), then the other two projections of this point are determined using auxiliary lines - a generatrix located on the surface of the cone and drawn through the point BUT , or a circle located in a plane parallel to the base of the cone.

Figure 140

In the first case (Figure 140, a) through the point A carry out a frontal projection 1""S"" auxiliary generatrix. Using a vertical line of communication drawn from the point 1 , located on the frontal projection of the base circle, find the horizontal projection 1" this generatrix, on which, with the help of a communication line passing through A" , find desired point A .

In the second case (Figure 140, b) an auxiliary line passing through the point BUT , there will be a circle located on a conical surface and parallel to the plane H - parallel. The frontal projection of this circle is depicted as a segment 1""1"" horizontal straight line, the value of which is equal to the diameter of the auxiliary circle. Desired horizontal projection A" points BUT is located at the intersection of the communication line, lowered from the point A" , with a horizontal projection of the auxiliary circle.

If a given frontal projection 1"" points 1 located on the contour (outline) generatrix, then the horizontal projection of the point is without auxiliary lines.

AT isometric view point BUT , located on the surface of the cone, is built in three coordinates (see Figure 140, c):  X ,  Y and Z BUT O along the axis X delayed coordinate  X  Y z Z BUT BUT .

Ball. A ball (Figure 141) is a body obtained by rotating a semicircle ABC (generating) around its diameter AC (axis of rotation), and the surface that the arc describes in this case ABC , is called spherical or spherical. A ball refers to bodies limited only by a surface of revolution.

Figure 141

Ball(spherical) surface is the locus of points equidistant from one point O called ball center. If the ball is cut by horizontal planes, then circles will be obtained in the section - parallels. The largest of the parallels has a diameter equal to the diameter of the ball. Such a circle is called equator. The circles obtained as a result of sections of the ball by planes passing through its axis of rotation are called meridians.

Construction of projections of the ball and points on its surface

The projections of the ball are shown in Figure 142, a. Horizontal and frontal projections - circles of radius equal to the radius of the sphere.

Figure 142

If point BUT located on spherical surface, then the auxiliary line 1"" 2"" , drawn through this point parallel to the axis Oh (parallel), is projected onto the horizontal projection plane by a circle. On the horizontal projection of the auxiliary circle, the desired horizontal projection is found using the communication line A" points BUT .

The value of the diameter of the auxiliary circle is equal to the frontal projection 1""2"" .

Axonometric image spheres (ball) is made in the form of a circle (Figure 142 b), the radius of which is geometrically defined as the distance from the center of the sphere to the projection of the equator (ellipse) along its major axis (perpendicular to Oz ).

In axonometric projection, a point BUT , located on the surface of the ball, is built according to three coordinates: X BUT ,Y BUT and Z BUT . These coordinates are sequentially plotted in directions parallel to the isometric axes. In the example under consideration, from the point O along the axis X delayed coordinate X BUT ; a straight line is drawn from its end parallel to the y axis, on which the coordinate is plotted Y BUT ; from the end of the segment, parallel to the axis z a straight line is drawn, on which the coordinate is plotted Z BUT . As a result of the constructions, we obtain the desired point BUT .

Thor- a body (Figure 143) formed by the rotation of a circle or its arc around an axis located in the same plane with it but not passing through the center of the circle or its arc.

Figure 143

If the axis of rotation does not intersect the generating circle, then the torus is called ring(open torus) (Figure 143, a). If the axis of rotation intersects the generating circle, then it turns out barrel-shaped torus(closed torus or intersecting torus) (Figure 143, b). In the latter case, the generatrix of the torus surface is the arc ABC circles.

The largest of the circles that describe the points of the generatrix of the torus surface is called equator, and the smallest throat, or neck.

Construction of torus projections

A circular ring (or an open torus) has a horizontal projection in the form of two concentric circles, the difference in radii of which is equal to the thickness of the ring or the diameter of the generating circle (Figure 145). The frontal projection is limited to the right and left by arcs of semicircles of the diameter of the generating circle.

Figure 144, a and b shows two types of a closed torus. In the first case, the generating arc of a circle of radius R away from the axis of rotation at a distance less than the radius R , and in the second case - more. In both cases, the frontal projections of the torus are a real view of two generating arcs of a circle of radius R located symmetrically with respect to the frontal projection of the axis of rotation. The profile projections of the torus will be circles.

Figure 144

Constructing points on the surface of a torus

In the case where the point BUT lies on the surface of a circular ring and one of its projections is given, to find the second projection of this point, an auxiliary circle is used passing through given point BUT and located on the surface of the ring in a plane perpendicular to the axis of the ring (Figure 145).

If frontal projection is set A"" points BUT lying on the surface of the ring, then to find its second projection (in this case, horizontal) through A" carry out a frontal projection of the auxiliary circle - a segment of a horizontal straight line 2""2"" . Then build a horizontal projection 2"2" this circle and on it, using a line of communication, find a point A" .

If the horizontal projection is given B" points B located on the surface of this ring, then to find the frontal projection of this point through 1" carry out a horizontal projection of the auxiliary circle of radius R 1 . Then through the left and right points 1" and 1" of this circle, vertical communication lines are drawn until they intersect with the frontal projections of the sketch generatrix of the circle of radius R and get points 1"" and 1"" . These points are connected by a horizontal straight line, which is a frontal projection of the auxiliary circle (it will be visible). Drawing a vertical line from a point B" to the intersection with the line 1""1"" get the desired point B"" .

The same construction techniques are applicable to points located on the surface of the torus.

Figure 145

Building an axonometric image The torus can be divided into three stages (Figure 146). First, a projection of the radial axial line (the trajectory of the center of the generating circle) is constructed in the form of an ellipse. Then we determine the radius of the sphere touching the torus along the generatrix (circle). To do this, we build the projection of the frontal sketch generatrix of the torus in the form of a smaller ellipse. The radius of the sphere is defined as the length of the segment O 1 F from the center of the ellipse to a point on that ellipse that lies on the major axis of the ellipse (perpendicular Oy ). Next, we build a large number of circles with a radius R spheres with centers on the projection of the radial axial torus O 1 … O n (the more, the more accurate the contour of the future torus). Finally, we draw the contour line of the torus as a line tangent to each circle of the sphere.

Figure 146

AT axonometric projection point BUT , located on the surface of the torus, is built according to three coordinates: X BUT ,Y BUT and Z BUT . These coordinates are sequentially plotted in directions parallel to the isometric axes.

The surfaces of revolution and the bodies bounded by them have wide application in many areas of technology: a cathode ray tube balloon (Fig. 8.11, a), center of the lathe (Fig. 8.11, b) volumetric microwave resonator electromagnetic oscillations(Fig. 8.11, in), storage dewar vessel liquid air(Fig. 8.11, G), electron collector of a powerful cathode-ray device (Fig. 8.11, e), etc.

Depending on the type of generatrix of the surface, rotations can be ruled, non-linear, or consist of parts of such surfaces.

A surface of revolution is a surface resulting from the rotation of a generatrix around a fixed line. straight-axis surfaces.

In the drawings, the axis is represented by a dash-dotted line. The generating line can general case have both curved and straight sections. The surface of revolution in the drawing can be specified by the generatrix and the position of the axis. Figure 8.12 shows the surface of revolution, which is formed by the rotation of the generatrix AlCD (her frontal projection a"b"c"d") around axis OO 1 (front projection o"o 1" , perpendicular to the plane N. During rotation, each point of the generatrix describes a circle, the plane of which is perpendicular to the axis. Accordingly, the line of intersection of the surface of revolution by any plane perpendicular to the axis is a circle. Such circles are called parallels. The top view (Fig. 8.12) shows projections of circles described by points A, B, C and D, passing through projections a, b, c, d. The largest parallel of the two parallels adjacent to it on both sides of it is called equator, likewise the smallest throat.

The plane passing through the axis of the surface of revolution is called meridional the line of its intersection with the surface of revolution - meridian. If the axis of the surface is parallel to the plane of projections, then the meridian lying in a plane parallel to this plane of projections is calledmain meridian.The main meridian is projected onto this plane of projections without distortion. So, if the axis of the surface of revolution is parallel to the plane V, then the prime meridian is projected onto the plane V without distortion, e.g. projection a"f"b"c"d". If the axis of the surface of revolution is perpendicular to the plane H, then the horizontal projection of the surface has an outline in the form of a circle.

The most convenient for performing images of surfaces of revolution are cases when their axes are perpendicular to the plane H, to the V plane or to the W plane.

Some surfaces of revolutionare special cases of the surfaces considered in 8.1, for example, a cylinder of revolution, a cone of revolution. For a cylinder and a cone of revolution, the meridians are straight lines. They are parallel to the axis and equidistant from it for a cylinder or intersect the axis at the same point at the same angle to the axis for a cone. A cylinder and a cone of revolution are surfaces that are infinite in the direction of their generators; therefore, in the images they are limited by some lines, for example, by lines of intersection of these surfaces with projection planes or by any of the parallels. It is known from solid geometry that a right circular cylinder and a right circular cone are bounded by a surface of revolution and planes perpendicular to the axis of the surface. The meridian of such a cylinder is a rectangle, the meridian of a cone is a triangle.

Such a surface of revolution as a sphere is limited and can be shown in full in the drawing. The equator and meridians of the sphere are equal circles. At orthogonal projection on all three projection planes, the outlines of the sphere are projected into a circle.

Thor. When a circle (or its arc) rotates around an axis that lies in the plane of this circle, but does not pass through its center, a surface called a torus is obtained. Figure 8.13 shows: an open torus, or a circular ring, - Figure 8.13, a, closed torus - figure 8.13, b, self-intersecting torus - figure 8.13, c, Tor (Fig. 8.13, d) also called lemon. In Figure 8.13 they are shown in a position where the axis of the torus is perpendicular to the plane of projections N. Spheres can be inscribed in open and closed tori. A torus can be viewed as a surface enveloping identical spheres whose centers are on a circle.

In the constructions on the drawings, two systems of circular sections of the torus are widely used: in planes perpendicular to its axis, and in planes passing through the axis of the torus. At the same time, in flat

In the directions perpendicular to the axis of the torus, in turn, there are two families of circles - lines of intersection of planes with the outer surface of the torus and lines of intersection of planes with the inner surface of the torus. The lemon-shaped torus (Fig. 8.13, d) has only the first family of circles.

In addition, the torus also has a third system of circular sections, which lie in planes passing through the center of the torus and tangent to it. inner surface. Figure 8.14 shows circular sections with centers o 1r and o 2r on an additional projection plane R, formed by the front-projecting plane Q(Qv), passing through the center of the torus with projections oh" oh and tangent to the inner surface of the torus at points with projections 1", 1, 2" 2. Point projections 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10make the drawing easier to read. Diameter d these circular sections equal to length major axes of ellipses into which circular sections are projected onto horizontal plane projections: d = 2R.

Points on a surface of revolution.The position of a point on the surface of revolution is determined by the belonging of the point to the line of the surface frame, i.e., with the help of a circle passing through this point on the surface of revolution. In the case of ruled surfaces, rectilinear generators can also be used for this purpose.

The use of a parallel and a rectilinear generatrix for constructing projections of points belonging to a given surface of revolution is shown in Figure 8.12. If a

given the projection t", then carry out a frontal projection f"f1" parallels and then radius R draw a circle - a horizontal projection of a parallel - and find a projection on it t. If a horizontal projection were given t, then it would be necessary to draw a radius R=om circle, build f "on the point f" and draw f"f1"- frontal projection of the parallel - and mark a point on it in the projection connection t". If given a projection P" on a ruled (conical) section of the surface of revolution, then a frontal projection is carried out d"s" sketch generatrix and through the projection n "- frontal projection s "to" generatrix on the surface of the cone. Then in plan view sk this generatrix constructs a projection n. If the horizontal projection n were given, then the horizontal projection should be drawn through it sk generatrix, by projection k" and s" (its construction was discussed above) build a frontal projection s"to" and on it in the projection connection mark the projection n "

Figure 8.15 shows the construction of point projections TO, belonging to the surface of the torus. It should be noted that the construction is made for visible horizontal projections to and front projection to".

Figure 8.16 shows the construction according to a given frontal projection t" points on the surface of a sphere of its horizontal t and profile t " projections. Projection t built using a circle - a parallel passing through the projection m". Its radius is o-1. Projection m "" is built using a circle, the plane of which is parallel to the profile plane of the projections passing through the projection t". Its radius is about "2".

The construction of projections of lines on the surface of revolution can also be performed using circles - parallels passing through the points belonging to this line.

Figure 8.17 shows the construction of a horizontal projection a line defined by frontal projection a"b" on a surface of revolution, consisting of parts of the surfaces of a sphere, torus, conic. For a more accurate drawing of the horizontal projection of the line, we continue its frontal projection up and down and mark the projections 6" and 5" extreme points. Horizontal projections 6, 1, 3, 4, 5 built with communication lines. Projections b, 2, 7, 8, and constructed using parallels whose frontal projections pass through the projections b"2", 7", 8", a" these points. Quantity and location intermediate points choose based on the shape of the line and the required accuracy of construction. Horizontal projection line consists of sections: b-1 - parts of the ellipse,

Examples of solids of revolution

• Ball - formed by a semicircle rotating around the diameter of the cut
• Cylinder - formed by a rectangle rotating around one of the sides

For the area of ​​the lateral surface of the cylinder, the area of ​​its development is taken: Sside = 2πrh.

• Cone - formed right triangle, rotating around one of the legs

The area of ​​its development is taken as the area of ​​the lateral surface of the cone: Sside = πrl Area full surface cones: Scon = πr(l+ r)

When the contours of figures are rotated, a surface of revolution arises (for example, a sphere formed by a circle), while when a filled contour rotates, bodies arise (like a ball formed by a circle).

Volume and surface area of ​​bodies of revolution

• The first Guldin-Papp theorem states:

• The second Guldin-Pappa theorem states:

Literature

A.V. Pogorelov. "Geometry. Grade 10-11» §21. Bodies of rotation. - 2011

Notes

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Books

• A set of tables. Mathematics. Polyhedra. bodies of revolution. 11 tables + 64 cards + methodology,. Educational album of 11 sheets (format 68 x 98 cm): - Parallel design. - Image of flat figures. - Step-by-step illustration of the proof of theorems. - Mutual arrangement of lines and ...
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