Misplaced Pages

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43-590: The surface area of Cheboksary Reservoir is 2,190 square kilometres (850 sq mi), max width is 16 kilometres (9.9 mi), max depth is 35 metres (115 ft). The reservoir has partly flooded the Mari Depression . The largest cities on the Reservoir are Nizhny Novgorod , Cheboksary and Kozmodemyansk . This Russian location article is a stub . You can help Misplaced Pages by expanding it . This Nizhny Novgorod Oblast location article

86-661: A b ∫ 0 2 π ‖ ∂ r ∂ t × ∂ r ∂ θ ‖   d θ   d t . {\displaystyle A_{x}=\iint _{S}dS=\iint _{[a,b]\times [0,2\pi ]}\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt.} Computing

129-431: A b ∫ 0 2 π ‖ y ⟨ y cos ⁡ ( θ ) d x d t , y sin ⁡ ( θ ) d x d t , y d y d t ⟩ ‖   d θ   d t = ∫

172-1367: A b ∫ 0 2 π y ( d x d t ) 2 + ( d y d t ) 2   d θ   d t = ∫ a b 2 π y ( d x d t ) 2 + ( d y d t ) 2   d t {\displaystyle {\begin{aligned}A_{x}&=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }\left\|y\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle \right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\cos ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\sin ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}2\pi y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ dt\end{aligned}}} where

215-467: A b ∫ 0 2 π y cos 2 ⁡ ( θ ) ( d x d t ) 2 + sin 2 ⁡ ( θ ) ( d x d t ) 2 + ( d y d t ) 2   d θ   d t = ∫

258-403: A b y ( t ) ( d x d t ) 2 + ( d y d t ) 2 d t . {\displaystyle A_{x}=2\pi \int _{a}^{b}y(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt.} If the continuous curve is described by the function y = f ( x ) ,

301-439: A cell imposes upper limits on size, as the volume increases much faster than does the surface area, thus limiting the rate at which substances diffuse from the interior across the cell membrane to interstitial spaces or to other cells. Indeed, representing a cell as an idealized sphere of radius r , the volume and surface area are, respectively, V = (4/3) πr and SA = 4 πr . The resulting surface area to volume ratio

344-598: A ≤ x ≤ b , then the integral becomes A x = 2 π ∫ a b y 1 + ( d y d x ) 2 d x = 2 π ∫ a b f ( x ) 1 + ( f ′ ( x ) ) 2 d x {\displaystyle A_{x}=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx=2\pi \int _{a}^{b}f(x){\sqrt {1+{\big (}f'(x){\big )}^{2}}}\,dx} for revolution around

387-536: A , b ] , and the axis of revolution is the y -axis, then the surface area A y is given by the integral A y = 2 π ∫ a b x ( t ) ( d x d t ) 2 + ( d y d t ) 2 d t , {\displaystyle A_{y}=2\pi \int _{a}^{b}x(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,} provided that x ( t )

430-696: A curve described by y = f ( x ) {\displaystyle y=f(x)} around the x-axis may be most simply described by y 2 + z 2 = f ( x ) 2 {\displaystyle y^{2}+z^{2}=f(x)^{2}} . This yields the parametrization in terms of x {\displaystyle x} and θ {\displaystyle \theta } as ( x , f ( x ) cos ⁡ ( θ ) , f ( x ) sin ⁡ ( θ ) ) {\displaystyle (x,f(x)\cos(\theta ),f(x)\sin(\theta ))} . If instead we revolve

473-564: A great deal of care. This should provide a function which assigns a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity : the area of the whole is the sum of the areas of the parts . More rigorously, if a surface S is a union of finitely many pieces S 1 , …, S r which do not overlap except at their boundaries, then Surface areas of flat polygonal shapes must agree with their geometrically defined area . Since surface area

SECTION 10

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516-758: A plane curve is given by ⟨ x ( t ) , y ( t ) ⟩ {\displaystyle \langle x(t),y(t)\rangle } then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by r ( t , θ ) = ⟨ y ( t ) cos ⁡ ( θ ) , y ( t ) sin ⁡ ( θ ) , x ( t ) ⟩ {\displaystyle \mathbf {r} (t,\theta )=\langle y(t)\cos(\theta ),y(t)\sin(\theta ),x(t)\rangle } with 0 ≤ θ ≤ 2 π {\displaystyle 0\leq \theta \leq 2\pi } . Then

559-434: Is a stub . You can help Misplaced Pages by expanding it . Surface area This is an accepted version of this page The surface area (symbol A ) of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of

602-451: Is a geometric notion, areas of congruent surfaces must be the same and the area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions . These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth . Such surfaces consist of finitely many pieces that can be represented in

645-410: Is credited to Archimedes . Surface area is important in chemical kinetics . Increasing the surface area of a substance generally increases the rate of a chemical reaction . For example, iron in a fine powder will combust , while in solid blocks it is stable enough to use in structures. For different applications a minimal or maximal surface area may be desired. The surface area of an organism

688-635: Is important in several considerations, such as regulation of body temperature and digestion . Animals use their teeth to grind food down into smaller particles, increasing the surface area available for digestion. The epithelial tissue lining the digestive tract contains microvilli , greatly increasing the area available for absorption. Elephants have large ears , allowing them to regulate their own body temperature. In other instances, animals will need to minimize surface area; for example, people will fold their arms over their chest when cold to minimize heat loss. The surface area to volume ratio (SA:V) of

731-399: Is never negative between the endpoints a and b . This formula is the calculus equivalent of Pappus's centroid theorem . The quantity ( d x d t ) 2 + ( d y d t ) 2 {\displaystyle {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}} comes from

774-481: Is that surface area cannot be defined simply as the limit of areas of polyhedral shapes approximating a given smooth surface. It was demonstrated by Hermann Schwarz that already for the cylinder, different choices of approximating flat surfaces can lead to different limiting values of the area; this example is known as the Schwarz lantern . Various approaches to a general definition of surface area were developed in

817-431: Is the surface of revolution of the curve between two given points which minimizes surface area . A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution. There are only two minimal surfaces of revolution ( surfaces of revolution which are also minimal surfaces): the plane and the catenoid . A surface of revolution given by rotating

860-613: Is therefore 3/ r . Thus, if a cell has a radius of 1 μm, the SA:V ratio is 3; whereas if the radius of the cell is instead 10 μm, then the SA:V ratio becomes 0.3. With a cell radius of 100, SA:V ratio is 0.03. Thus, the surface area falls off steeply with increasing volume. Surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix ) one full revolution around an axis of rotation (normally not intersecting

903-469: The Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity 2π x ( t ) is the path of (the centroid of) this small segment, as required by Pappus' theorem. Likewise, when the axis of rotation is the x -axis and provided that y ( t ) is never negative, the area is given by A x = 2 π ∫

SECTION 20

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946-1018: The cross product yields ∂ r ∂ t × ∂ r ∂ θ = ⟨ y cos ⁡ ( θ ) d x d t , y sin ⁡ ( θ ) d x d t , y d y d t ⟩ = y ⟨ cos ⁡ ( θ ) d x d t , sin ⁡ ( θ ) d x d t , d y d t ⟩ {\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}=\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle =y\left\langle \cos(\theta ){\frac {dx}{dt}},\sin(\theta ){\frac {dx}{dt}},{\frac {dy}{dt}}\right\rangle } where

989-479: The parametric form with a continuously differentiable function r → . {\displaystyle {\vec {r}}.} The area of an individual piece is defined by the formula Thus the area of S D is obtained by integrating the length of the normal vector r → u × r → v {\displaystyle {\vec {r}}_{u}\times {\vec {r}}_{v}} to

1032-459: The x -axis, and A y = 2 π ∫ a b x 1 + ( d y d x ) 2 d x {\displaystyle A_{y}=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx} for revolution around the y -axis (provided a ≥ 0 ). These come from the above formula. This can also be derived from multivariable integration. If

1075-1109: The case of the spherical curve with radius r , y ( x ) = √ r − x rotated about the x -axis A = 2 π ∫ − r r r 2 − x 2 1 + x 2 r 2 − x 2 d x = 2 π r ∫ − r r r 2 − x 2 1 r 2 − x 2 d x = 2 π r ∫ − r r d x = 4 π r 2 {\displaystyle {\begin{aligned}A&=2\pi \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,dx\\&=2\pi r\int _{-r}^{r}\,{\sqrt {r^{2}-x^{2}}}\,{\sqrt {\frac {1}{r^{2}-x^{2}}}}\,dx\\&=2\pi r\int _{-r}^{r}\,dx\\&=4\pi r^{2}\,\end{aligned}}} A minimal surface of revolution

1118-399: The circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus ). The sections of the surface of revolution made by planes through the axis are called meridional sections . Any meridional section can be considered to be the generatrix in the plane determined by it and the axis. The sections of

1161-654: The cone r → u {\displaystyle {\vec {r}}_{u}} = partial derivative of r → {\displaystyle {\vec {r}}} with respect to u {\displaystyle u} , r → v {\displaystyle {\vec {r}}_{v}} = partial derivative of r → {\displaystyle {\vec {r}}} with respect to v {\displaystyle v} , D {\displaystyle D} = shadow region The below given formulas can be used to show that

1204-448: The curve around the y-axis, then the curve is described by y = f ( x 2 + z 2 ) {\displaystyle y=f({\sqrt {x^{2}+z^{2}}})} , yielding the expression ( x cos ⁡ ( θ ) , f ( x ) , x sin ⁡ ( θ ) ) {\displaystyle (x\cos(\theta ),f(x),x\sin(\theta ))} in terms of

1247-413: The generatrix, except at its endpoints). The volume bounded by the surface created by this revolution is the solid of revolution . Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated around any diameter generates a sphere of which it is then a great circle , and if

1290-410: The late nineteenth and the early twentieth century by Henri Lebesgue and Hermann Minkowski . While for piecewise smooth surfaces there is a unique natural notion of surface area, if a surface is very irregular, or rough, then it may not be possible to assign an area to it at all. A typical example is given by a surface with spikes spread throughout in a dense fashion. Many surfaces of this type occur in

1333-546: The parameters x {\displaystyle x} and θ {\displaystyle \theta } . If x and y are defined in terms of a parameter t {\displaystyle t} , then we obtain a parametrization in terms of t {\displaystyle t} and θ {\displaystyle \theta } . If x {\displaystyle x} and y {\displaystyle y} are functions of t {\displaystyle t} , then

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1376-889: The partial derivatives yields ∂ r ∂ t = ⟨ d y d t cos ⁡ ( θ ) , d y d t sin ⁡ ( θ ) , d x d t ⟩ , {\displaystyle {\frac {\partial \mathbf {r} }{\partial t}}=\left\langle {\frac {dy}{dt}}\cos(\theta ),{\frac {dy}{dt}}\sin(\theta ),{\frac {dx}{dt}}\right\rangle ,} ∂ r ∂ θ = ⟨ − y sin ⁡ ( θ ) , y cos ⁡ ( θ ) , 0 ⟩ {\displaystyle {\frac {\partial \mathbf {r} }{\partial \theta }}=\langle -y\sin(\theta ),y\cos(\theta ),0\rangle } and computing

1419-1032: The same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar. For example, the spherical surface with unit radius is generated by the curve y ( t ) = sin( t ) , x ( t ) = cos( t ) , when t ranges over [0,π] . Its area is therefore A = 2 π ∫ 0 π sin ⁡ ( t ) ( cos ⁡ ( t ) ) 2 + ( sin ⁡ ( t ) ) 2 d t = 2 π ∫ 0 π sin ⁡ ( t ) d t = 4 π . {\displaystyle {\begin{aligned}A&{}=2\pi \int _{0}^{\pi }\sin(t){\sqrt {{\big (}\cos(t){\big )}^{2}+{\big (}\sin(t){\big )}^{2}}}\,dt\\&{}=2\pi \int _{0}^{\pi }\sin(t)\,dt\\&{}=4\pi .\end{aligned}}} For

1462-460: The study of fractals . Extensions of the notion of area which partially fulfill its function and may be defined even for very badly irregular surfaces are studied in geometric measure theory . A specific example of such an extension is the Minkowski content of the surface. r = Internal radius, h = height s = slant height of the cone, r = radius of the circular base, h = height of

1505-438: The surface area for polyhedra (i.e., objects with flat polygonal faces ), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere , are assigned surface area using their representation as parametric surfaces . This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration . A general definition of surface area

1548-453: The surface area is given by the surface integral A x = ∬ S d S = ∬ [ a , b ] × [ 0 , 2 π ] ‖ ∂ r ∂ t × ∂ r ∂ θ ‖   d θ   d t = ∫

1591-865: The surface area of a sphere and cylinder of the same radius and height are in the ratio 2 : 3 , as follows. Let the radius be r and the height be h (which is 2 r for the sphere). Sphere surface area = 4 π r 2 = ( 2 π r 2 ) × 2 Cylinder surface area = 2 π r ( h + r ) = 2 π r ( 2 r + r ) = ( 2 π r 2 ) × 3 {\displaystyle {\begin{array}{rlll}{\text{Sphere surface area}}&=4\pi r^{2}&&=(2\pi r^{2})\times 2\\{\text{Cylinder surface area}}&=2\pi r(h+r)&=2\pi r(2r+r)&=(2\pi r^{2})\times 3\end{array}}} The discovery of this ratio

1634-455: The surface of revolution made by planes that are perpendicular to the axis are circles. Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular. If the curve is described by the parametric functions x ( t ) , y ( t ) , with t ranging over some interval [

1677-409: The surface of revolution obtained by revolving the curve around the x-axis is described by ( x ( t ) , y ( t ) cos ⁡ ( θ ) , y ( t ) sin ⁡ ( θ ) ) {\displaystyle (x(t),y(t)\cos(\theta ),y(t)\sin(\theta ))} , and the surface of revolution obtained by revolving the curve around

1720-449: The surface over the appropriate region D in the parametric uv plane. The area of the whole surface is then obtained by adding together the areas of the pieces, using additivity of surface area. The main formula can be specialized to different classes of surfaces, giving, in particular, formulas for areas of graphs z = f ( x , y ) and surfaces of revolution . One of the subtleties of surface area, as compared to arc length of curves,

1763-649: The trigonometric identity sin 2 ⁡ ( θ ) + cos 2 ⁡ ( θ ) = 1 {\displaystyle \sin ^{2}(\theta )+\cos ^{2}(\theta )=1} was used. With this cross product, we get A x = ∫ a b ∫ 0 2 π ‖ ∂ r ∂ t × ∂ r ∂ θ ‖   d θ   d t = ∫

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1806-425: The y-axis is described by ( x ( t ) cos ⁡ ( θ ) , y ( t ) , x ( t ) sin ⁡ ( θ ) ) {\displaystyle (x(t)\cos(\theta ),y(t),x(t)\sin(\theta ))} . Meridians are always geodesics on a surface of revolution. Other geodesics are governed by Clairaut's relation . A surface of revolution with

1849-434: Was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory , which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface. While the areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires

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