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75-484: A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology . The shift in the basic meaning—solid versus surface (as in a solid ball versus sphere surface)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces . In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object,

150-416: A = b , with radius r = a = b . For the parabola, the standard form has the focus on the x -axis at the point ( a , 0) and the directrix the line with equation x = − a . In standard form the parabola will always pass through the origin. For a rectangular or equilateral hyperbola, one whose asymptotes are perpendicular, there is an alternative standard form in which the asymptotes are

225-401: A . (Here a is the semi-major axis defined below.) A parabola may also be defined in terms of its focus and latus rectum line (parallel to the directrix and passing through the focus): it is the locus of points whose distance to the focus plus or minus the distance to the line is equal to 2 a ; plus if the point is between the directrix and the latus rectum, minus otherwise. In addition to

300-487: A Dedekind-infinite set , having a proper subset equinumerous to itself. If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set , then it has many well-orderings which are non-isomorphic. Important ideas discussed by David Burton in his book The History of Mathematics: An Introduction include how to define "elements" or parts of

375-574: A plane , called the cutting plane , with the surface of a double cone (a cone with two nappes ). It is usually assumed that the cone is a right circular cone for the purpose of easy description, but this is not required; any double cone with some circular cross-section will suffice. Planes that pass through the vertex of the cone will intersect the cone in a point, a line or a pair of intersecting lines. These are called degenerate conics and some authors do not consider them to be conics at all. Unless otherwise stated, "conic" in this article will refer to

450-412: A change of coordinates ( rotation and translation of axes ) these equations can be put into standard forms . For ellipses and hyperbolas a standard form has the x -axis as principal axis and the origin (0,0) as center. The vertices are (± a , 0) and the foci (± c , 0) . Define b by the equations c = a − b for an ellipse and c = a + b for a hyperbola. For a circle, c = 0 so

525-454: A closed curve tangent to the line at infinity. Further extension, by expanding the real coordinates to admit complex coordinates, provides the means to see this unification algebraically. The conic sections have been studied for thousands of years and have provided a rich source of interesting and beautiful results in Euclidean geometry . A conic is the curve obtained as the intersection of

600-424: A common integration technique for finding volumes of solids of revolution. In the treatise by this name, written c.  225 BCE , Archimedes obtained the result of which he was most proud, namely obtaining the formulas for the volume and surface area of a sphere by exploiting the relationship between a sphere and its circumscribed right circular cylinder of the same height and diameter . The sphere has

675-437: A countable infinite subset. If a set of sets is infinite or contains an infinite element, then its union is infinite. The power set of an infinite set is infinite. Any superset of an infinite set is infinite. If an infinite set is partitioned into finitely many subsets, then at least one of them must be infinite. Any set which can be mapped onto an infinite set is infinite. The Cartesian product of an infinite set and

750-426: A cylinder always means a circular cylinder. The height (or altitude) of a cylinder is the perpendicular distance between its bases. The cylinder obtained by rotating a line segment about a fixed line that it is parallel to is a cylinder of revolution . A cylinder of revolution is a right circular cylinder. The height of a cylinder of revolution is the length of the generating line segment. The line that

825-558: A cylindric section that is an ellipse, the eccentricity e of the cylindric section and semi-major axis a of the cylindric section depend on the radius of the cylinder r and the angle α between the secant plane and cylinder axis, in the following way: e = cos ⁡ α , a = r sin ⁡ α . {\displaystyle {\begin{aligned}e&=\cos \alpha ,\\[1ex]a&={\frac {r}{\sin \alpha }}.\end{aligned}}} If

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900-406: A focus; its half-length is the semi-latus rectum ( ℓ ). The focal parameter ( p ) is the distance from a focus to the corresponding directrix. The major axis is the chord between the two vertices: the longest chord of an ellipse, the shortest chord between the branches of a hyperbola. Its half-length is the semi-major axis ( a ). When an ellipse or hyperbola are in standard position as in

975-470: A line, five points determine a conic . Formally, given any five points in the plane in general linear position , meaning no three collinear , there is a unique conic passing through them, which will be non-degenerate; this is true in both the Euclidean plane and its extension, the real projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear

1050-506: A measure of how far the ellipse deviates from being circular. If the angle between the surface of the cone and its axis is β {\displaystyle \beta } and the angle between the cutting plane and the axis is α , {\displaystyle \alpha ,} the eccentricity is cos ⁡ α cos ⁡ β . {\displaystyle {\frac {\cos \alpha }{\cos \beta }}.} A proof that

1125-400: A non-degenerate conic. There are three types of conics: the ellipse , parabola , and hyperbola . The circle is a special kind of ellipse, although historically Apollonius considered it a fourth type. Ellipses arise when the intersection of the cone and plane is a closed curve . The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone; for

1200-401: A nonempty set is infinite. The Cartesian product of an infinite number of sets, each containing at least two elements, is either empty or infinite; if the axiom of choice holds, then it is infinite. If an infinite set is a well-ordered set , then it must have a nonempty, nontrivial subset that has no greatest element. In ZF, a set is infinite if and only if the power set of its power set is

1275-405: A plane not parallel to the given line. Such cylinders have, at times, been referred to as generalized cylinders . Through each point of a generalized cylinder there passes a unique line that is contained in the cylinder. Thus, this definition may be rephrased to say that a cylinder is any ruled surface spanned by a one-parameter family of parallel lines. A cylinder having a right section that

1350-416: A polyhedral viewpoint, a cylinder can also be seen as a dual of a bicone as an infinite-sided bipyramid . Infinite set In set theory , an infinite set is a set that is not a finite set . Infinite sets may be countable or uncountable . The set of natural numbers (whose existence is postulated by the axiom of infinity ) is infinite. It is the only set that is directly required by

1425-401: A right circular cylinder, there are several ways in which planes can meet a cylinder. First, planes that intersect a base in at most one point. A plane is tangent to the cylinder if it meets the cylinder in a single element. The right sections are circles and all other planes intersect the cylindrical surface in an ellipse . If a plane intersects a base of the cylinder in exactly two points then

1500-411: A right cone, this means the cutting plane is perpendicular to the axis. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola . In the remaining case, the figure is a hyperbola : the plane intersects both halves of the cone, producing two separate unbounded curves. Compare also spheric section (intersection of a plane with

1575-401: A right cylinder, is more generally given by L = e × p , {\displaystyle L=e\times p,} where e is the length of an element and p is the perimeter of a right section of the cylinder. This produces the previous formula for lateral area when the cylinder is a right circular cylinder. A right circular hollow cylinder (or cylindrical shell ) is

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1650-1233: A set, how to define unique elements in the set, and how to prove infinity. Burton also discusses proofs for different types of infinity, including countable and uncountable sets. Topics used when comparing infinite and finite sets include ordered sets , cardinality, equivalency, coordinate planes , universal sets , mapping, subsets, continuity, and transcendence . Cantor's set ideas were influenced by trigonometry and irrational numbers. Other key ideas in infinite set theory mentioned by Burton, Paula, Narli and Rodger include real numbers such as π , integers, and Euler's number . Both Burton and Rogers use finite sets to start to explain infinite sets using proof concepts such as mapping, proof by induction, or proof by contradiction. Mathematical trees can also be used to understand infinite sets. Burton also discusses proofs of infinite sets including ideas such as unions and subsets. In Chapter 12 of The History of Mathematics: An Introduction , Burton emphasizes how mathematicians such as Zermelo , Dedekind , Galileo , Kronecker , Cantor, and Bolzano investigated and influenced infinite set theory. Many of these mathematicians either debated infinity or otherwise added to

1725-423: A short and wide disk cylinder has a diameter much greater than its height. A cylindric section is the intersection of a cylinder's surface with a plane . They are, in general, curves and are special types of plane sections . The cylindric section by a plane that contains two elements of a cylinder is a parallelogram . Such a cylindric section of a right cylinder is a rectangle . A cylindric section in which

1800-408: A single real line (actually a coincident pair of lines), or only at the vertex. These cases give rise to the hyperbolic, parabolic or elliptic cylinders respectively. This concept is useful when considering degenerate conics , which may include the cylindrical conics. A solid circular cylinder can be seen as the limiting case of a n -gonal prism where n approaches infinity . The connection

1875-542: A single real point.) If A and B have different signs and ρ ≠ 0 {\displaystyle \rho \neq 0} , we obtain the hyperbolic cylinders , whose equations may be rewritten as: ( x a ) 2 − ( y b ) 2 = 1. {\displaystyle \left({\frac {x}{a}}\right)^{2}-\left({\frac {y}{b}}\right)^{2}=1.} Finally, if AB = 0 assume, without loss of generality , that B = 0 and A = 1 to obtain

1950-410: A sphere, producing a circle or point), and spherical conic (intersection of an elliptic cone with a concentric sphere). Alternatively, one can define a conic section purely in terms of plane geometry: it is the locus of all points P whose distance to a fixed point F (called the focus ) is a constant multiple e (called the eccentricity ) of the distance from P to a fixed line L (called

2025-610: A three-dimensional region bounded by two right circular cylinders having the same axis and two parallel annular bases perpendicular to the cylinders' common axis, as in the diagram. Let the height be h , internal radius r , and external radius R . The volume is given by V = π ( R 2 − r 2 ) h = 2 π ( R + r 2 ) h ( R − r ) . {\displaystyle V=\pi \left(R^{2}-r^{2}\right)h=2\pi \left({\frac {R+r}{2}}\right)h(R-r).} Thus,

2100-409: A volume two-thirds that of the circumscribed cylinder and a surface area two-thirds that of the cylinder (including the bases). Since the values for the cylinder were already known, he obtained, for the first time, the corresponding values for the sphere. The volume of a sphere of radius r is ⁠ 4 / 3 ⁠ π r = ⁠ 2 / 3 ⁠ (2 π r ) . The surface area of this sphere

2175-419: Is 4 π r = ⁠ 2 / 3 ⁠ (6 π r ) . A sculpted sphere and cylinder were placed on the tomb of Archimedes at his request. In some areas of geometry and topology the term cylinder refers to what has been called a cylindrical surface . A cylinder is defined as a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in

2250-429: Is a curve obtained from a cone's surface intersecting a plane . The three types of conic section are the hyperbola , the parabola , and the ellipse ; the circle is a special case of the ellipse, though it was sometimes called as a fourth type. The ancient Greek mathematicians studied conic sections, culminating around 200 BC with Apollonius of Perga 's systematic work on their properties. The conic sections in

2325-756: Is a generalization of the equation of the ordinary, circular cylinder ( a = b ). Elliptic cylinders are also known as cylindroids , but that name is ambiguous, as it can also refer to the Plücker conoid . If ρ {\displaystyle \rho } has a different sign than the coefficients, we obtain the imaginary elliptic cylinders : ( x a ) 2 + ( y b ) 2 = − 1 , {\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=-1,} which have no real points on them. ( ρ = 0 {\displaystyle \rho =0} gives

Cylinder - Misplaced Pages Continue

2400-619: Is a specialization of the homogeneous form used in the more general setting of projective geometry (see below ). The conic sections described by this equation can be classified in terms of the value B 2 − 4 A C {\displaystyle B^{2}-4AC} , called the discriminant of the equation. Thus, the discriminant is − 4Δ where Δ is the matrix determinant | A B / 2 B / 2 C | . {\displaystyle \left|{\begin{matrix}A&B/2\\B/2&C\end{matrix}}\right|.} If

2475-399: Is again the determinant of the 2 × 2 matrix. In the case of an ellipse the squares of the two semi-axes are given by the denominators in the canonical form. In polar coordinates , a conic section with one focus at the origin and, if any, the other at a negative value (for an ellipse) or a positive value (for a hyperbola) on the x -axis, is given by the equation where e is

2550-406: Is also a countably infinite set, even if it is a proper subset of the integers. The set of all rational numbers is a countably infinite set as there is a bijection to the set of integers. The set of all real numbers is an uncountably infinite set. The set of all irrational numbers is also an uncountably infinite set. Conic section A conic section , conic or a quadratic curve

2625-401: Is also called the semi-minor axis. The following relations hold: For conics in standard position, these parameters have the following values, taking a , b > 0 {\displaystyle a,b>0} . After introducing Cartesian coordinates , the focus-directrix property can be used to produce the equations satisfied by the points of the conic section. By means of

2700-637: Is an ellipse , parabola , or hyperbola is called an elliptic cylinder , parabolic cylinder and hyperbolic cylinder , respectively. These are degenerate quadric surfaces . When the principal axes of a quadric are aligned with the reference frame (always possible for a quadric), a general equation of the quadric in three dimensions is given by f ( x , y , z ) = A x 2 + B y 2 + C z 2 + D x + E y + G z + H = 0 , {\displaystyle f(x,y,z)=Ax^{2}+By^{2}+Cz^{2}+Dx+Ey+Gz+H=0,} with

2775-424: Is the diameter of the circular top or bottom. For a given volume, the right circular cylinder with the smallest surface area has h = 2 r . Equivalently, for a given surface area, the right circular cylinder with the largest volume has h = 2 r , that is, the cylinder fits snugly in a cube of side length = altitude ( = diameter of base circle). The lateral area, L , of a circular cylinder, which need not be

2850-635: Is the equation of an elliptic cylinder . Further simplification can be obtained by translation of axes and scalar multiplication. If ρ {\displaystyle \rho } has the same sign as the coefficients A and B , then the equation of an elliptic cylinder may be rewritten in Cartesian coordinates as: ( x a ) 2 + ( y b ) 2 = 1. {\displaystyle \left({\frac {x}{a}}\right)^{2}+\left({\frac {y}{b}}\right)^{2}=1.} This equation of an elliptic cylinder

2925-426: Is the only type of geometric figure for which this technique works with the use of only elementary considerations (no appeal to calculus or more advanced mathematics). Terminology about prisms and cylinders is identical. Thus, for example, since a truncated prism is a prism whose bases do not lie in parallel planes, a solid cylinder whose bases do not lie in parallel planes would be called a truncated cylinder . From

3000-408: Is the product of the area of a base and the height. For example, an elliptic cylinder with a base having semi-major axis a , semi-minor axis b and height h has a volume V = Ah , where A is the area of the base ellipse (= π ab ). This result for right elliptic cylinders can also be obtained by integration, where the axis of the cylinder is taken as the positive x -axis and A ( x ) = A

3075-406: Is very strong and many older texts treat prisms and cylinders simultaneously. Formulas for surface area and volume are derived from the corresponding formulas for prisms by using inscribed and circumscribed prisms and then letting the number of sides of the prism increase without bound. One reason for the early emphasis (and sometimes exclusive treatment) on circular cylinders is that a circular base

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3150-414: The right circular cylinder . The definitions and results in this section are taken from the 1913 text Plane and Solid Geometry by George A. Wentworth and David Eugene Smith ( Wentworth & Smith 1913 ). A cylindrical surface is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in a plane not parallel to

3225-1489: The Cartesian coordinate system , the graph of a quadratic equation in two variables is always a conic section (though it may be degenerate ), and all conic sections arise in this way. The most general equation is of the form with all coefficients real numbers and A, B, C not all zero. The above equation can be written in matrix notation as ( x y ) ( A B / 2 B / 2 C ) ( x y ) + ( D E ) ( x y ) + F = 0. {\displaystyle {\begin{pmatrix}x&y\end{pmatrix}}{\begin{pmatrix}A&B/2\\B/2&C\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+{\begin{pmatrix}D&E\end{pmatrix}}{\begin{pmatrix}x\\y\end{pmatrix}}+F=0.} The general equation can also be written as ( x y 1 ) ( A B / 2 D / 2 B / 2 C E / 2 D / 2 E / 2 F ) ( x y 1 ) = 0. {\displaystyle {\begin{pmatrix}x&y&1\end{pmatrix}}{\begin{pmatrix}A&B/2&D/2\\B/2&C&E/2\\D/2&E/2&F\end{pmatrix}}{\begin{pmatrix}x\\y\\1\end{pmatrix}}=0.} This form

3300-405: The Euclidean plane have various distinguishing properties, many of which can be used as alternative definitions. One such property defines a non-circular conic to be the set of those points whose distances to some particular point, called a focus , and some particular line, called a directrix , are in a fixed ratio, called the eccentricity . The type of conic is determined by the value of

3375-472: The axioms to be infinite. The existence of any other infinite set can be proved in Zermelo–Fraenkel set theory (ZFC), but only by showing that it follows from the existence of the natural numbers. A set is infinite if and only if for every natural number, the set has a subset whose cardinality is that natural number. If the axiom of choice holds, then a set is infinite if and only if it includes

3450-449: The directrix ). For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. A circle is a limiting case and is not defined by a focus and directrix in the Euclidean plane. The eccentricity of a circle is defined to be zero and its focus is the center of the circle, but its directrix can only be taken as the line at infinity in the projective plane. The eccentricity of an ellipse can be seen as

3525-408: The parabolic cylinders with equations that can be written as: x 2 + 2 a y = 0. {\displaystyle x^{2}+2ay=0.} In projective geometry , a cylinder is simply a cone whose apex (vertex) lies on the plane at infinity . If the cone is a quadratic cone, the plane at infinity (which passes through the vertex) can intersect the cone at two real lines,

3600-489: The surface area of a right circular cylinder, oriented so that its axis is vertical, consists of three parts: The area of the top and bottom bases is the same, and is called the base area , B . The area of the side is known as the lateral area , L . An open cylinder does not include either top or bottom elements, and therefore has surface area (lateral area) L = 2 π r h {\displaystyle L=2\pi rh} The surface area of

3675-426: The volume of a right circular cylinder have been known from early antiquity. A right circular cylinder can also be thought of as the solid of revolution generated by rotating a rectangle about one of its sides. These cylinders are used in an integration technique (the "disk method") for obtaining volumes of solids of revolution. A tall and thin needle cylinder has a height much greater than its diameter, whereas

3750-429: The 2 × 2 matrix) are invariant under arbitrary rotations and translations of the coordinate axes, as is the determinant of the 3 × 3 matrix above . The constant term F and the sum D + E are invariant under rotation only. When the conic section is written algebraically as the eccentricity can be written as a function of the coefficients of the quadratic equation. If 4 AC = B

3825-402: The above curves defined by the focus-directrix property are the same as those obtained by planes intersecting a cone is facilitated by the use of Dandelin spheres . Alternatively, an ellipse can be defined in terms of two focus points, as the locus of points for which the sum of the distances to the two foci is 2 a ; while a hyperbola is the locus for which the difference of distances is 2

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3900-453: The area of each elliptic cross-section, thus: V = ∫ 0 h A ( x ) d x = ∫ 0 h π a b d x = π a b ∫ 0 h d x = π a b h . {\displaystyle V=\int _{0}^{h}A(x)dx=\int _{0}^{h}\pi abdx=\pi ab\int _{0}^{h}dx=\pi abh.} Using cylindrical coordinates ,

3975-423: The base of a circular cylinder has a radius r and the cylinder has height h , then its volume is given by V = π r 2 h {\displaystyle V=\pi r^{2}h} This formula holds whether or not the cylinder is a right cylinder. This formula may be established by using Cavalieri's principle . In more generality, by the same principle, the volume of any cylinder

4050-903: The coefficients being real numbers and not all of A , B and C being 0. If at least one variable does not appear in the equation, then the quadric is degenerate. If one variable is missing, we may assume by an appropriate rotation of axes that the variable z does not appear and the general equation of this type of degenerate quadric can be written as A ( x + D 2 A ) 2 + B ( y + E 2 B ) 2 = ρ , {\displaystyle A\left(x+{\frac {D}{2A}}\right)^{2}+B\left(y+{\frac {E}{2B}}\right)^{2}=\rho ,} where ρ = − H + D 2 4 A + E 2 4 B . {\displaystyle \rho =-H+{\frac {D^{2}}{4A}}+{\frac {E^{2}}{4B}}.} If AB > 0 this

4125-424: The conic can be deduced from its equation. In the Euclidean plane, the three types of conic sections appear quite different, but share many properties. By extending the Euclidean plane to include a line at infinity, obtaining a projective plane , the apparent difference vanishes: the branches of a hyperbola meet in two points at infinity, making it a single closed curve; and the two ends of a parabola meet to make it

4200-401: The conic is non-degenerate , then: In the notation used here, A and B are polynomial coefficients, in contrast to some sources that denote the semimajor and semiminor axes as A and B . The discriminant B – 4 AC of the conic section's quadratic equation (or equivalently the determinant AC – B /4 of the 2 × 2 matrix) and the quantity A + C (the trace of

4275-411: The conic is a parabola and its eccentricity equals 1 (provided it is non-degenerate). Otherwise, assuming the equation represents either a non-degenerate hyperbola or ellipse, the eccentricity is given by where η = 1 if the determinant of the 3 × 3 matrix above is negative and η = −1 if that determinant is positive. It can also be shown that the eccentricity is a positive solution of

4350-476: The coordinate axes and the line x = y is the principal axis. The foci then have coordinates ( c , c ) and (− c , − c ) . The first four of these forms are symmetric about both the x -axis and y -axis (for the circle, ellipse and hyperbola), or about the x -axis only (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines y = x and y = − x . These standard forms can be written parametrically as, In

4425-432: The eccentricity ( e ), foci, and directrix, various geometric features and lengths are associated with a conic section. The principal axis is the line joining the foci of an ellipse or hyperbola, and its midpoint is the curve's center . A parabola has no center. The linear eccentricity ( c ) is the distance between the center and a focus. The latus rectum is the chord parallel to the directrix and passing through

4500-446: The eccentricity and l is the semi-latus rectum. As above, for e = 0 , the graph is a circle, for 0 < e < 1 the graph is an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. The polar form of the equation of a conic is often used in dynamics ; for instance, determining the orbits of objects revolving about the Sun. Just as two (distinct) points determine

4575-453: The eccentricity. In analytic geometry , a conic may be defined as a plane algebraic curve of degree 2; that is, as the set of points whose coordinates satisfy a quadratic equation in two variables which can be written in the form A x 2 + B x y + C y 2 + D x + E y + F = 0. {\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0.} The geometric properties of

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4650-419: The equation can be converted to canonical form in transformed variables x ~ , y ~ {\displaystyle {\tilde {x}},{\tilde {y}}} as or equivalently where λ 1 {\displaystyle \lambda _{1}} and λ 2 {\displaystyle \lambda _{2}} are the eigenvalues of

4725-410: The equation where again Δ = A C − B 2 4 . {\displaystyle \Delta =AC-{\frac {B^{2}}{4}}.} This has precisely one positive solution—the eccentricity— in the case of a parabola or ellipse, while in the case of a hyperbola it has two positive solutions, one of which is the eccentricity. In the case of an ellipse or hyperbola,

4800-402: The equations below, with foci on the x -axis and center at the origin, the vertices of the conic have coordinates (− a , 0) and ( a , 0) , with a non-negative. The minor axis is the shortest diameter of an ellipse, and its half-length is the semi-minor axis ( b ), the same value b as in the standard equation below. By analogy, for a hyperbola the parameter b in the standard equation

4875-415: The generatrix is an element of the cylindrical surface. A solid bounded by a cylindrical surface and two parallel planes is called a (solid) cylinder . The line segments determined by an element of the cylindrical surface between the two parallel planes is called an element of the cylinder . All the elements of a cylinder have equal lengths. The region bounded by the cylindrical surface in either of

4950-403: The given line. Any line in this family of parallel lines is called an element of the cylindrical surface. From a kinematics point of view, given a plane curve, called the directrix , a cylindrical surface is that surface traced out by a line, called the generatrix , not in the plane of the directrix, moving parallel to itself and always passing through the directrix. Any particular position of

5025-407: The ideas of infinite sets. Potential historical influences, such as how Prussia's history in the 1800s, resulted in an increase in scholarly mathematical knowledge, including Cantor's theory of infinite sets. One potential application of infinite set theory is in genetics and biology. The set of all integers , {..., −1, 0, 1, 2, ...} is a countably infinite set. The set of all even integers

5100-409: The intersecting plane intersects and is perpendicular to all the elements of the cylinder is called a right section . If a right section of a cylinder is a circle then the cylinder is a circular cylinder. In more generality, if a right section of a cylinder is a conic section (parabola, ellipse, hyperbola) then the solid cylinder is said to be parabolic, elliptic and hyperbolic, respectively. For

5175-402: The line segment joining these points is part of the cylindric section. If such a plane contains two elements, it has a rectangle as a cylindric section, otherwise the sides of the cylindric section are portions of an ellipse. Finally, if a plane contains more than two points of a base, it contains the entire base and the cylindric section is a circle. In the case of a right circular cylinder with

5250-498: The matrix ( A B / 2 B / 2 C ) {\displaystyle \left({\begin{matrix}A&B/2\\B/2&C\end{matrix}}\right)} — that is, the solutions of the equation — and S {\displaystyle S} is the determinant of the 3 × 3 matrix above , and Δ = λ 1 λ 2 {\displaystyle \Delta =\lambda _{1}\lambda _{2}}

5325-433: The parallel planes is called a base of the cylinder. The two bases of a cylinder are congruent figures. If the elements of the cylinder are perpendicular to the planes containing the bases, the cylinder is a right cylinder , otherwise it is called an oblique cylinder . If the bases are disks (regions whose boundary is a circle ) the cylinder is called a circular cylinder . In some elementary treatments,

5400-403: The segment is revolved about is called the axis of the cylinder and it passes through the centers of the two bases. The bare term cylinder often refers to a solid cylinder with circular ends perpendicular to the axis, that is, a right circular cylinder, as shown in the figure. The cylindrical surface without the ends is called an open cylinder . The formulae for the surface area and

5475-438: The solid right circular cylinder is made up the sum of all three components: top, bottom and side. Its surface area is therefore A = L + 2 B = 2 π r h + 2 π r 2 = 2 π r ( h + r ) = π d ( r + h ) {\displaystyle A=L+2B=2\pi rh+2\pi r^{2}=2\pi r(h+r)=\pi d(r+h)} where d = 2 r

5550-476: The volume of a cylindrical shell equals 2 π  ×   average radius ×   altitude ×  thickness. The surface area, including the top and bottom, is given by A = 2 π ( R + r ) h + 2 π ( R 2 − r 2 ) . {\displaystyle A=2\pi \left(R+r\right)h+2\pi \left(R^{2}-r^{2}\right).} Cylindrical shells are used in

5625-554: The volume of a right circular cylinder can be calculated by integration V = ∫ 0 h ∫ 0 2 π ∫ 0 r s d s d ϕ d z = π r 2 h . {\displaystyle {\begin{aligned}V&=\int _{0}^{h}\int _{0}^{2\pi }\int _{0}^{r}s\,\,ds\,d\phi \,dz\\[5mu]&=\pi \,r^{2}\,h.\end{aligned}}} Having radius r and altitude (height) h ,

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