What is Surface area: Definition and 450 Discussions

The surface area 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 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 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.

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  1. M

    Surface Area Vector in Exterior Algebra 3D

    Hello every one . What is the Surface Area vector form in exterior algebra ,I mean by that the Surface Area vector as an exterior form in 3D , just like the volume form .THANKS
  2. avito009

    Why does surface area of an Event Horizon increase?

    Stephen hawking came to know of a study that stated that surface area of an event horizon increases. So he said that since area has increased Entropy also had increased. But why does surface area of an Event Horizon increase?
  3. K

    Calculate area of subsection of a three-dimensional surface

    For the following three-dimensional surface, z = -4.53 + 2.67x + 2.78y - 1.09xy, I would like to calculate the area for each of three subsections of this surface: (1) for which z is in between the corresponding x and y values (i.e., x < z < y OR y < z < x); (2) for which x is in between the...
  4. M

    Calculate Circle Radius/Diameter from Surface Area

    Hi How is circular surface area values square rooted ??. I am using this formula r2Xπ=mm2 to calculate the surface area of a circle but i want to know if there is a formula to get back to the diameter or the radius of the circle ?. example i have a circular surface area of 5.26mm2 and i want...
  5. K

    Relationship between surface area and damping coefficient

    I have been investigating the damping of an oscillating mass on a spring by a piece of card of varying surface area, in order to work out the damping constant λ for the equation y = A₀ e-λt sin(ωt). I took the natural log of the amplitude of each successive oscillation and plotting that against...
  6. C

    Conceptual problem regarding pressure and surface area

    Is the air pressure acting on the top of the head of a person standing in a room equal to the air pressure acting on a same area on the floor of the same room? Since F = pA , it seems as if the air pressure is independent of the height of the surface on which it is acting. But that doesn't seem...
  7. gracy

    Understanding Surface Area: Foldings, Invaginations, Roughness & Powder

    why surface area increases with foldings ,invaginations,roughness and powdered form?please someone explain concept of surface area. what is difference between perimeter or (circumference in case of circle)and surface area? Mentor Note: Two posts have been merged.
  8. M

    Surface Area of Cylinder inside of sphere

    Homework Statement Find the area of that part of the cylinder x^2 + y^2 = 2ay that lies inside the sphere x^2 + y^2 + z^2 = 4a^2. Homework Equations [/B] If a surface S can be parametrized in terms of two variables u and v, then dS = Norm[dR(u,v)/du x dR(u,v)/dv]. The surface area is given...
  9. I

    Surface plasmon dependence on surface area to volume ratio

    Hey everyone, I was recently reading a paper on surface enhanced Raman scattering, and it mentioned that plasmons (and for that matter surface plasmon polaritons-where my interest lies) are sensitive to the surface to volume ratio of the structure. I can begin to understand intuitively with...
  10. G

    Spherical shape surface area calculation

    Homework Statement Homework Equations [/B] Surface area of sphere = 4*Pi*r^2, where r: is the radius of the sphere circleThe Attempt at a Solution Solution:[/B] 1. In terms of “r” and “R”, and the radius of sphere “S”, and “d”: Given that: · Surface area of both shaded area are equal...
  11. O

    What is the difference between a sphere and a ball?

    I'm trying to follow Schwabl Thermodynamics, and I found the following equation for the surface area of a unit d-sphere: $$ \int d\Omega_d = \frac{2 \pi^{d/2}}{\Gamma(d/2)} $$ But this formula clearly fails for d=1: should be $$\pi$$ and d=2: should be $$ 4 \pi $$. What gives?
  12. S

    Surface Area and electric field strength

    This is my first post but I have frequented the forum for a little while now. I tried to figure things out myself and often times I am lead here by google. So my question is this: How is the electric field of an object altered when the surface area of the object is altered? Example: Take a...
  13. ChrisVer

    Area of a Sphere: Intuitive Problem Solved

    I am sure this should have already be discussed somewhere in the past... I have an intuitive problem with the area of a sphere. Following the mathematics of the metric and surfaces, I can easily derive the area of a sphere which is 4 \pi R^{2} . Now I'm have this problem: Suppose I get a ring...
  14. shintashi

    Surface area of a Hypersphere?

    I'm wondering how to scale up the surface area of a sphere of 6 million meters in radius, into a hypersphere of similar radius (i.e. a Hyper Earth). I would also like to know the ratio. I would like to know the basic value in 4th dimension, but knowing values for 5th, 6th, and higher would...
  15. J

    Show rectangular box of given volume has minimum surface area when

    Homework Statement show rectangular box of given volume has minimum surface area when the box is a cube [gotta show it with partial derivatives to minimize] Homework Equations surface area = 2(wl+hl+hw) volume = whl The Attempt at a Solution so this is the one I would be minimizing...
  16. A

    MHB Quick Surface Area and Application Question

    For the first one, I'm completely dumbfounded. I used the SA formula for and ended up getting 2 $π\int_{π/6}^{0} \, cos2x * √1+4sin(2x)^2 ) $ And took u = sin2x, ultimately giving me $ π\int_{√3/2}^{0} \,d (1+ 4u^2)^(1/2) $ I'm lost as to what to do next. I'm almost definite that my...
  17. S

    Surface area of between 2 cones

    find the area of the cylinder x^2+z^2=a^2 that is inside the cylinder x^2+y^2=a^2. my attempt: parameterise x^2+z^2=a^2 as a vector r(x,y) = (x,y,(a^2-x^2)^1/2). using the formula given here : http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceArea.aspx, I found the surface area = the...
  18. I

    MHB Finding Surface Area from Equation in Terms of x=g(y)

    if I am finding the surface area, and I am given the equation in term of x=g(y) about the x-axis, do i have to solve for y or can i just integrate in terms of y? would i just use dx/dy instead?
  19. P

    Rotational frictions relationship to surface area

    Hi, I have a question I'd like some help with which is related to my job working with drill rigs. Where you have a cylinder such as a drill string where the forces of the ground pressing in from all sides how much influence does the surface area have on the rotational force required to...
  20. Nathanael

    Surface area of the Sphere

    I'm wondering why my method for finding the surface area of a sphere is invalid. Essentially I'm integrating the perimeter of the circle perpendicular to the radius along the entire radius, and then multiplying by 2 (because the radius only covers half the sphere) (I hope that made sense...
  21. K

    A spherical raindrops evaporates at rate proportional to surface area?

    i want to find V(t) At first i found this problem was very simple but when i try to write differential equations i ended up with these V' = kA that's for sure then i confined the problem only to spherical shape and no other shapes of raindrops involved as i can't express A in term of V alone(...
  22. J

    Surface area and volume uniquely determine a shape

    Is this so? I cannot think of a counter-example and it is too general a statement to prove.
  23. D

    MHB Double Integral Problem/ Surface Area of parametric surface

    Hi I'm doing a surface area problem with a parametric surface and I got the cross product but I can't figure out the double integral. I found the solution online but with no explanation, so can someone explain how to solve this integral: thank you!
  24. M

    Surface area problem involving solids of revolution

    Homework Statement Evaluate the definite integral for the surface area generated by revolving the curve about the y-axis: Homework Equations Curve: y=9-x^2 about y-axis The Attempt at a Solution Attached
  25. L

    Surface Area rotated about an axis which is not the x or y axis

    Hi. I understand how to solve surface Area using integration when it is to be revolved about the x or y axis. But when the axis is not x or y I have a difficult time solving it. Please help me. Here is the equation sqrt(x+1) rotated at x=-1 and y=5. the bounds are 1 to 5. since y=sqrt(x+1)...
  26. I

    Differentials of spherical surface area and volume

    please tell me if i did this correctly: task: I'm trying to divide the differential dA by dV where.. dA = differential surface area of a sphere, dV = differential volume of a sphere dA=8\pirdr dV=4\pir2dr so then dA/dV= 2/r Also, if i treat this as a derivative, then would...
  27. P

    Increasing Wing Surface Area with Pores?

    Hello, I'm new here, but I have some questions which i'd like to ask. I had this idea about corrugated wings to increase the wing surface area (thus increasing lift), but quickly realized that this would also increase the wings drag as it would have a larger cross section breaking through...
  28. D

    MHB Surface Area Formula for \(z = x^2 + y^2\) Derivation

    I would like to derive the surface area for an equation in the form of \(z = f(x, y)\). For example, if I have a sphere \((b^2 = x^2 + y^2)\), the surface area is circumference times arc length \((SA = 2\pi r\ell)\). Here I can take an arc and break it up into n parts to find the differential...
  29. binbagsss

    Spherical surface area element

    See image attached. (I've had a google but can't find anything). I am trying to understand the expression : Rdθ.2∏Rsinθ Here are my thoughts so far: Rdθ is the width of a strip, θ being the variable changing/to integrate over, giving arise to the elements. 2∏Rsinθ must then...
  30. D

    MHB Surface Area of ball floating in water

    Using calculus, how would I derive a formula for the exposed surface area of a ball floating in water? For such a formula to be a good candidate, it would have to consider oscillations of the water and placid water. The surface area of a sphere is \(A = 4\pi r^2\).
  31. A

    MHB Stuck trying to integrate the surface area of a curve

    Here's the problem I was given: Find the area of the surface generated by revolving the curve x=\frac{e^y + e^{-y} }{2} from 0 \leq y \leq ln(2) about the y-axis. I tried the normal route first... g(y) = x = \frac{1}{2} (e^y + e^{-y}) g'(y) = dx/dy = \frac{1}{2} (e^y - e^{-y}) S = \int...
  32. aleksbooker

    Can't integrate the surface area of revolving curve the normal way

    Homework Statement Find the area of the surface generated by revolving the curve x=\frac{e^y + e^{-y} }{2} from 0 \leq y \leq ln(2) about the y-axis. The Attempt at a Solution I tried the normal route first... g(y) = x = \frac{1}{2} (e^y + e^{-y}) g'(y) = dx/dy = \frac{1}{2}...
  33. Philosophaie

    Geodesic Equation & Orbital Surface Area Around the Sun

    The "s" in the geodesic equation refers to the "surface area" for that portion of the orbit around a star or black hole. For a small enough "delta t" the surface areas are the same. Around a small star the orbital surface area (without the other interfering gravitational sources) would...
  34. S

    Luminosity and Surface Area

    Homework Statement A typical adult burns about 2500 Calories in one day. 1)How much energy does the average human emit every second 2)What is another term to use for expressing a “Joule per second”? 3)From your answers to (1) and (2), you just determined the “luminosity” of the average human...
  35. I

    Deriving the surface area equation

    consider the following image (the red is the surface area element and the green is the differential element that I'm integrating over) when we derived this in class, we treated the area formed by vectors a and b, as the area of a parallelogram. the thing is, a and b should be at right...
  36. C

    What is the surface area when a curve is rotated about the x-axis?

    Homework Statement Obtain the surface area when the curve y=ex, 0≤x≤1, is rotated about the x-axis Homework Equations Surface Area = 2∏a∫b x√(1+(dy/dx)2)dx The Attempt at a Solution I started with the the equation, Surface Area = 2∏0∫1 x√(1+e2x)dx. However, whichever way I try to...
  37. C

    Surface area from bands: Calculus

    Homework Statement The shaded band shown here is cut from a sphere of radius Rby parallel planes hunits apart. Show that the surface area of the band is 2piRh. The image is on this site: http://imgur.com/TCx1weD http://imgur.com/TCx1weD The Attempt at a Solution How do I do...
  38. MarkFL

    MHB Kuba's question at Yahoo Answers regarding finding the surface area of a torus

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  39. MarkFL

    MHB Bari's question at Yahoo Answers regarding minimizing the surface area of a silo

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  40. M

    Surface Area of Solid: f(x)=1/x, [1,∞) about x-axis

    Homework Statement f(x) = 1/x Interval [1, ∞) about the x-axis Set-up the integral for the surface area of the solid Then use the substitution u = x2 and integrate using the formula: ∫ sqrt(u2 + a2) / u2 du = ln(u + sqrt(u2 + a2) - sqrt(u2 + a2) / u + C a is a constant Homework...
  41. T

    MHB Surface area limited with two functions

    I have to calculate the surface area limited with two functions: g(x)=2 and h(x)=\frac{4}{4x^2+1} i was thinking first to calculate the points where this functions are intersecting by g(x)=h(x) and to find corresponding $x$ for the intersection...and then to calculate definite integral...
  42. Philosophaie

    Double Integral Surface Area of Spherical Ball

    Homework Statement Double Integral Surface Area of Spherical Ball radius Homework Equations ##\int_S d\vec{S} = 4*\pi*a^2## The Attempt at a Solution ##\int\int_0^a f(r,?) dr d? = 4*\pi*a^2##
  43. Philosophaie

    Surface Area and Volume of a Sphere

    I need to know the Surface Area and Volume of a spherical ball at the origin radius a. What I want is to evaluate the integrals at each integral. ##\oint_S dS =\int\int d? d? = 4 *\pi*r^2## ##\oint_V dV = \int_0^{\pi}\int_0^{2\pi}\int_0^a dr d\theta d\phi## = ##\frac{4}{3}*\pi*a^2##
  44. karush

    MHB How fast is the surface area increasing

    The volume of a cube is increasing at a rate of $$\displaystyle\frac{10 cm^3}{\text {min}}$$ How fast is the surface area increasing when the length of an edge is $$\displaystyle 30 \text{ cm}$$ The answer to this is $$\displaystyle \frac{4}{3} \text{ cm}^2$$ $$V=x^3$$ and $$SA=6x^2$$ and...
  45. E

    Surface area of a bulb filament

    Does anyone know what it is, or where I can find the surface area of a typical 60W incandescent bulb filament? (if possible, by GE) I'm working on a research project and we're applying the stefan-boltzmann law. I'm calculating an area around 3e-5 m^2. I figure I'm in the ballpark, but I'd...
  46. J

    Surface area of Revolution

    Homework Statement I want the surface area of Revolution about x from [-2,2] of y = |x| So I want to know if I can take it x from [0,2] and just multiply this result by 2? Homework Equations The Attempt at a Solution Set up dy/dx = (5/(2√x)) S = 2 ∏ ∫ 5x^1/2(√ 1...
  47. D

    Question about surface area/ volume

    Homework Statement We have two animals. One of them is as big as an elephant or it's an elephant and the other one is a small animal as a cat. When we drop them from a high place, the smaller animal hurts less compared to the bigger one. Why? Homework Equations Surface area / volume The Attempt...
  48. PsychonautQQ

    Parametric equations to find surface area

    Homework Statement Which of the following integrals represents the area of the surface obtained by rotating the parametric curve x=t-t^2 y=(4/3)t^(3/2) 1<t<2 Homework Equations A = integral ( 2pi(y) * sqrt( 1+ (dy/dx)^2))dx The Attempt at a Solution I solved for dy/dx and got...
  49. K

    Surface Area frictional force

    Homework Statement does surface area to affect the values of coefficient µk? Homework Equations frictional force =µk * normal force The Attempt at a Solution I know that if mass is larger then the frictional force is higher( f= ma) but how about the surface area? does it affect...
  50. J

    Surface Area of Cube & Inscribed Sphere

    Is there any relationship between the surface area of a cube and the surface area of the cube's inscribed sphere? Jean~
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