Cavalieri's principle
The principle of Cavalieri (also known as the set of the Cavalieri principle or Cavalierisches ) is a statement from the geometry, which goes back to the Italian mathematician Bonaventura Cavalieri.
Principle of Cavalieri
The principle of Cavalieri says:
Another formulation is:
Classification and history
In modern analytical approach on geometry and measure theory, the principle of Cavalieri is a special case of the set of Fubini. Cavalieri himself had not a rigorous proof of principle, but took advantage of it to justify his method of indivisibles, which he presented to Geometria indivisibilibus and 1647 Exercitationes Geometricae 1635. This he was able to calculate the volume for some body and go beyond the results of Archimedes and Kepler. The idea, due calculating volumes on land was an important step in the development of integral calculus dar.
From the principle of Cavalieri can be deduced that ' höhengedehnten ' the volume of a body ( at constant area) is proportional to its height. As an example: A body whose height is doubled in this way, can be constructed by two same initial body by first all equivalent cut surfaces are merged and they are stacked in the order corresponding to the output body ( both initial body are more or less pushed into each other ).
Application Examples
Cylinder
The sections of a cylinder with planes perpendicular to the rotation axis are circular disks with surface area when the radius of the base respectively. According to the principle of Cavalieri is the volume of the cylinder is equal to the same height of a cuboid whose base has the same surface area, thus for example, the edge lengths and has. The volume of the cylinder is thus.
Hemisphere
The section of a half- sphere of radius with a plane which is in height parallel to the base, according to the Pythagorean theorem is a circle with a radius
The surface area of the cut surface is thus
The reference block in this example is a cylinder with the same base and height as the hemisphere from which a substance on the top circular cone was cut out. The sectional area in the height is an annulus with an outer radius and inner radius of the area is thus also
So the two bodies meet the principle of Cavalieri and therefore have the same volume. The volume of the reference block is the difference of the volumes of the cylinder and plug, so
Doubling delivers the well-known formula for the spherical volume.
Related to integral calculus
The idea behind the principle of Cavalieri found many again in the integral calculus. An example of smaller dimensions by one, so the lengths of the sections of straight lines with two faces, the equation
Is that essentially says that the area between the function graphs of and just as great as the area under the function graph of the difference; this latter area is precisely characterized by the fact that their vertical cuts of the same length as the sections of.
In the modern theoretical approach to the relationship between integral and area or volume but is typically made differently; the principle of Cavalieri is less important.
Reference to measure theory
The set of Cavalieri in the elemental form described above is a special case of the general theorem:
Be measurable. Then also, and
And it is true for almost all x and y, measurable ( or over ) or where the k-dimensional Lebesgue measure (volume) call. In particular: Is also measurable and is valid for almost all x, then. The same applies for and.