Body of revolution

Body of revolution is in the geometry of a body called the surface thereof by rotation of a generating curve around an axis of rotation is formed (see surface of revolution ). The axis of rotation is also called the figure axis . The curve lies in one plane and the axis lies in the same plane. A well-known solid of revolution is the torus . It is formed by rotating a circle . Also cones and cylinders are rotating body.

The volume and the surface are calculated using the so-called Guldin's rules (named after the mathematician and astronomer Paul Guldin ). Already in ancient times these were known as barycentric rules or centrobaric rules and were described by the Greek mathematician Pappos of Alexandria .

Representation of the rotation of a sine curve

Calculation of the volume of a solid of revolution

If the generating curve intersects the axis of rotation, it must be considered whether the corresponding partial volumes should be counted as positive or negative contributions to the total volume.

Rotation around the x axis

For a solid of revolution that results from the rotation of the area bounded by the graph of the function in the interval , the -axis and the two straight lines and around the -axis, the formula for calculating the volume is: ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle x}$ ${\displaystyle x=a}$ ${\displaystyle x=b}$ ${\displaystyle x}$

${\displaystyle V=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}$

Rotation around the y- axis

1st case: "disc integration"

Disc integration

When rotating (around the -axis) the area, which is limited by the graph of the function in the interval , the -axis and the two straight lines and , one has to transform to the inverse function . This exists when is continuous and strictly monotonic . If not (as e.g. in the picture above right), it can perhaps be broken down into sections, each of which is continuous and strictly monotonic. The volumes belonging to these sections must then be calculated and added separately. ${\displaystyle y}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle y}$ ${\displaystyle y=f(a)}$ ${\displaystyle y=f(b)}$ ${\displaystyle y=f(x)}$ ${\displaystyle x=f^{-1}(y)}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$

${\displaystyle V=\pi \cdot \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}(f^{-1}(y))^{2}\mathrm {d} y}$

If you substitute here , you get the volume around the axis ${\displaystyle x=f^{-1}(y)}$ ${\displaystyle y}$

${\displaystyle V=\pi \cdot \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}x^{2}\mathrm {d} y=\pi \cdot \int _{a}^{b}x^{2}\cdot \left|f'(x)\right|\mathrm {d} x}$.

The absolute value of and the min / max functions in the integral limits ensure a positive integral. ${\displaystyle f'}$

2nd case: "shell integration" (cylinder method)

Shell integration

For rotation (around the -axis) of the area, which is limited by the graph of the function in the interval , the -axis and the two straight lines and , the formula applies: ${\displaystyle y}$ ${\displaystyle f}$ ${\displaystyle [a,b]}$ ${\displaystyle x}$ ${\displaystyle x=a}$ ${\displaystyle x=b}$

${\displaystyle V=2\pi \,\int _{a}^{b}x\,f(x)\,\mathrm {d} x}$

Guldinian rules

The two Guldinian rules, named after the Swiss mathematician Paul Guldin , enormously shorten the surface and volume calculations of bodies of revolution , if the center of gravity of the lines or areas of the rotating objects can be easily identified using the symmetries of the respective task (see examples below).

Designations:

${\displaystyle M}$ = Surface

${\displaystyle V}$ = Volume

${\displaystyle L}$ = Length of the generating line (profile line)

${\displaystyle A}$ = Area of ​​the generating area

${\displaystyle R}$ = Radius of the center of gravity circle

${\displaystyle r}$ = Radius of the rotating circle (torus examples)

First rule

The area of a lateral surface of a solid of revolution whose axis of rotation does not intersect the generating line is equal to the product of the length of the generating line (profile line) and the circumference of the circle (center of gravity circle), which is generated by the rotation of the center of gravity of the profile line: ${\displaystyle M}$

${\displaystyle M=L\cdot 2\pi R}$

Expressed as a function of the function of the generating line, the area results as: ${\displaystyle f}$

When rotating around the x axis

${\displaystyle M=2\pi \int _{a}^{b}f(x){\sqrt {1+\left[f'(x)\right]^{2}}}\mathrm {d} x}$

With as coordinate of the line center of gravity of the line and its line element one finds ${\displaystyle \textstyle R=y_{s}={\frac {1}{L}}\int _{L}y\mathrm {d} L}$ ${\displaystyle y}$ ${\displaystyle L}$ ${\displaystyle \mathrm {d} L}$

${\displaystyle M=L\cdot 2\pi R=L\cdot 2\pi \cdot {\frac {1}{L}}\int _{L}f(x)\mathrm {d} L=2\pi \int _{L}f(x)\mathrm {d} L}$,

what the above result represents if the interval limits are still used. ${\displaystyle \textstyle \mathrm {d} L={\sqrt {(\mathrm {d} x)^{2}+(\mathrm {d} y)^{2}}}={\sqrt {1+\left({\frac {\mathrm {d} y}{\mathrm {d} x}}\right)^{2}}}\mathrm {d} x}$ ${\displaystyle x}$ ${\displaystyle [a,b]}$

When rotating around the y- axis

${\displaystyle M=2\pi \int _{\min(f(a),f(b))}^{\max(f(a),f(b))}f^{-1}(y){\sqrt {1+\left[\left(f^{-1}(y)\right)'\right]^{2}}}\mathrm {d} y}$

As with the volume calculation above, the calculation for the continuous and strictly monotonous sections of in which the inverse function exists must also be carried out separately. ${\displaystyle f(x)}$

Example: Surface of a rotational torus :

${\displaystyle M=2\pi r\cdot 2\pi R=4\pi ^{2}rR}$

See also: lateral surface

Second rule

The volume of a solid of revolution is equal to the product of the area of ​​the generating surface and the circumference of the circle, which is generated by the rotation of the center of gravity of this surface:

${\displaystyle V=A\cdot 2\pi R}$

In the following, the rotation of a surface around the -axis is considered; the case of a tilted rotation axis can be achieved by coordinate transformation. In the case of the rotation around the -axis of a surface between , the -axis and the boundaries and , the volume is expressed by using as the centroid ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle f(x)}$ ${\displaystyle x}$ ${\displaystyle x=a}$ ${\displaystyle x=b}$ ${\displaystyle f(x)}$ ${\displaystyle R}$

${\displaystyle V=A\cdot 2\pi {\tfrac {1}{A}}\int _{A}y\mathrm {d} A=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}$

with and . ${\displaystyle y={\tfrac {f(x)}{2}}}$ ${\displaystyle \mathrm {d} A=f(x)\mathrm {d} x}$

Example: Volume of a rotational torus:

${\displaystyle V=\pi r^{2}\cdot 2\pi R=2\pi ^{2}r^{2}R}$

Parametric shape

If a curve is defined by its parametric shape in an interval , the volumes of the bodies created by rotating the curve about the x-axis or the y-axis are given by ${\displaystyle (x(t),y(t))}$ ${\displaystyle [a,b]}$

${\displaystyle V_{x}=\int _{a}^{b}\pi y^{2}\,{\frac {\mathrm {d} x}{\mathrm {d} t}}\,\mathrm {d} t}$

${\displaystyle V_{y}=\int _{a}^{b}\pi x^{2}\,{\frac {\mathrm {d} y}{\mathrm {d} t}}\,\mathrm {d} t}$

The surface area of these bodies is given by

${\displaystyle M_{x}=\int _{a}^{b}2\pi y\,{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}}}\,\mathrm {d} t}$

${\displaystyle M_{y}=\int _{a}^{b}2\pi x\,{\sqrt {\left({\frac {\mathrm {d} x}{\mathrm {d} t}}\right)^{2}+\left({\frac {\mathrm {d} y}{\mathrm {d} t}}\right)^{2}}}\,\mathrm {d} t}$

Kepler's barrel rule

The Kepler barrel rule gives

${\displaystyle V={\frac {h}{6}}\cdot \left(q(0)+4q\left({\frac {h}{2}}\right)+q(h)\right)}$

as an approximation for the volume of a body whose cross-sectional area is known in three places. If the body is a body of revolution, the following applies for rotation around the axis: ${\displaystyle x}$

${\displaystyle V=\pi \cdot \int _{a}^{b}(f(x))^{2}\mathrm {d} x}$

${\displaystyle \approx \pi {\frac {b-a}{6}}\cdot \left((r(a))^{2}+4\left(r\left({\frac {a+b}{2}}\right)\right)^{2}+(r(b))^{2}\right)}$

For certain rotational body like sphere , cone , truncated cone , cylinder , paraboloid , hyperboloid of revolution and spheroid these are Formula , the exact volume of.

See also

• Surface of revolution
• Bullet
• cone
• Truncated cone
• cylinder
• Paraboloid of revolution
• Rotational hyperboloid
• Ellipsoid of revolution

Individual evidence

1. ↑ Kurt Magnus: Kreisel . Theory and applications. Springer, Berlin, Heidelberg 1971, ISBN 978-3-642-52163-8 , pp. 44 .
2. ^ Ilja N. Bronstein, Konstantin A. Semendjaew: Taschenbuch der Mathematik . 20th edition. Teubner; Nauka, Leipzig; Moscow 1981, p. 369 f . (XII, 860).
3. ↑ A. K. Sharma: Application Of Integral Calculus . Discovery Publishing House, 2005, ISBN 81-7141-967-4 , p. 168.
4. ^ Ravish R. Singh: Engineering Mathematics , 6th. Edition, Tata McGraw-Hill, 1993, ISBN 0-07-014615-2 , p. 6.90.

Web links

• Literature on solids of revolution in the catalog of the German National Library
• Ronny Harbich: body of revolution. ( Memento from March 15, 2011 in the Internet Archive ). At: Uni-Magdeburg.de. 2003 (PDF; 948 kB).