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. DryRun

    Find surface area of cylinder

    Homework Statement Find the surface area of the portion of the cylinder x^2+y^2=4y lying inside the sphere x^2+y^2+z^2=16Homework Equations Double integral for finding surface area. \iint_S d\sigma The Attempt at a Solution At first, i thought it was just the area of the surface of sphere...
  2. D

    Find surface area of a non-conducting sheet

    Homework Statement A small sphere of mass m carries a charge of q. It hangs from a silk thread which makes an angle θ with a large charged non-conducting sheet. Calculate the surface charge density on the sheet. Homework Equations ∫E.ds The Attempt at a Solution I found E of...
  3. ElijahRockers

    What is the surface area of a triangular region with a specific equation?

    Homework Statement Find the area of the surface z=7+2x+2y2 that lies above the triangle with vertices (0,0) (0,1) and (2,1) The Attempt at a Solution It's not a particularly difficult problem to set up, I just can't seem to get a simple integrand. ∫∫√(16y2+5) dA is what I come up...
  4. T

    Ductwork surface area calculation problem

    Hello, I have a major in economics and I am creating an excel spreadsheet for quotations of sheet metal ductwork as my final thesis. I need to calculate weights for various types of ductwork produced in the company I chose for my thesis. In order to calculate weight, I need to know the...
  5. R

    Surface area of a Planet based on gravatational acceleration and mass

    Realm of the Mad God actually takes place on a spherical planet where Oryx's minions thrive. This planet has a mass of 2.73 x 10^24 kg and a gravitational acceleration of 11.2 m/s^2 at its surface. Roughly 840,000 minions live per square kilometer. Together, the players of the realm kill 1500...
  6. J

    Surface Area of Shoe Box Shape to Maximize Volume

    Homework Statement Imagine you have a rectangular piece of cardboard measuring 3 feet by 4 feet. You know that if you cut a square out of each corner, you can fold the pieces together and tape them together to make an object that looks like a shoe...
  7. S

    Newton's third law and surface area of paddles

    why do paddles/oars have a large surface area? I thought that it does not really matter as i would just apply a force on the water so as to attain a action-reaction force on the boat thus pushing me forward. but if i look at the free body diagram of the paddle 1 force will be me pushing and the...
  8. L

    Why does increased surface area of a semi-conductor lead to better properties?

    I'm researching solar cells, some of which use TiO2 as a semiconducting film. Modern solar cells use TiO2 nanoparticles to increase the surface area of the film. Somehow this is advantageous to the solar cell. From my limited understanding of semiconductors, I know that to get a current...
  9. J

    Why Do Sphere Surface Areas Differ With Same Approximation?

    i have found the area of sphere in two ways using the same approximation.but i get two different answers ;one the correct value 4∏R^2?how does this happen? i'm attaching the solution below?please refer to the attachments and give a solution?
  10. A

    Surface Area of Hollowed Hemisphere - Basic Geometry

    I need to find the total surface area of the hollowed out hemisphere (picture attached), with an inner diameter of 1.86 cm and what looks like an outer radius (also what I am assuming as height) of 1.25 cm. Surface area of a sphere is 4∏r2, so half the sphere is 2∏r2. Since the outer...
  11. Z

    Finding Surface Area of Sphere with Illuminated Light Question

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  12. N

    Surface area of x = ln(y) - y^2/8 from 1 to e

    Homework Statement Find the area of the surface obtained from rotating the curve x = ln(y) - y^2/8 on [1,e] about the y-axis. Homework Equations SA = \int 2\pi*(f(x))*\sqrt{1+[f'(x)]^2}dx The Attempt at a Solution SA = \int 2\pi*(ln(y) - \frac{y^2}{8}))*\sqrt{1+\frac{9}{16*y^2}}dy from 1 to e...
  13. L

    Surface area obtained by solid of revolution

    Homework Statement Find surface area of solid of revolution obtained by rotating the curve: y=x2/40-5lnx from x=5 to x=7, rotated about x=-4 The Attempt at a Solution The problem is I know how to do this if I rotated it about x-axis/y-axis, but I have no idea how to do it if the...
  14. E

    Quick help surface area of a spherical cap

    i just need some help on how to start the process of proving it. suggestions/recommendations/anything will help!
  15. E

    Surface area of sin(x) rotated about the x-axis

    got stuck on doing the substitution.. any suggestions?
  16. P

    Surface area of the upper hemisphere

    Homework Statement For upper hemisphere S: x^2 + y^2 + z^2 = 1 , with z≥0, find the area element dS and unit normal vector N. Compute the total area of the hemisphere, ∫∫dS over S. Homework Equations Unit normal to surface f(x,y,z) = const N(hat) = grad(f)/|grad(f)| Surface area...
  17. B

    Calculating Cone Surface Area | Step-by-Step Guide

    Homework Statement I understand everything except for the part where the -1 pops out of nowhere on step 4. why? how?
  18. B

    Surface area of a beaker of water w/ surface tension?

    So I'm working on a chemistry lab determining Avogadro's constant with a fatty acid monolayer on a beaker of water, and my calculations are all work out to around 1.3*10^23 (ie a sixth of what they should be). The lab said "fill the beaker with water until it spills over". In doing so, I ended...
  19. S

    Calculate volume and surface area of a boiler

    Homework Statement A boiler drum is equipped with hemispherical heads (i.e. the two heads together make a sphere.) The diameter of the drum is 0.8 meters and the drum is 5 meters in length. Calculate: a.) The area of insulation required to cover the entire drum. b.) The total...
  20. N

    Finding surface area of cone in spherical coordinates

    Hello everyone, I recently tried to find the surface area of a hollow cone (there is no base, like an ice cream cone) using spherical coordinates. With cylindrical coordinates I was able to do this easily using the following integral: \int \int \frac{R}{h}z \sqrt{\frac{R^{2}}{h^{2}} + 1}...
  21. M

    Animal Brain size, mass, and surface area

    Animal Brain size, body mass, and surface area S.J. Gould wrote a series of interesting articles (chapters 22-24 in Ever Since Darwin) about the ratio of body mass to brain mass among animal species. Bigger animals have bigger brains (of course) but the ratio is getting smaller: a human's brain...
  22. DryRun

    Evalute surface area of top hemisphere

    Evaluate surface area of top hemisphere Homework Statement http://s1.ipicture.ru/uploads/20120106/jrHc5122.jpg The attempt at a solution This problem is new to me, in the sense that the integration is to be done against S, which is the same region S, over which the limits are defined...
  23. T

    Finding the surface area of one volume contained within another

    Homework Statement This isn't actually a homework problem, but rather a class of problems I'm running into as I study for prelims. I'm taking these from Greenspan's Calculus: An Introduction to Applied Mathematics. This type of problem has come up in the context of both volume and surface...
  24. C

    Surface area using double integrals

    Find the surface area of the triangle with vertices (0,0) (L, L) (L,-L) I know I have to take the double integrals of f(x,y) but I have no idea what f(x,y) is supposed to be!
  25. S

    Finding surface area using double integrals

    Homework Statement The portion of the paraboloid 2z=x^2+y^2 that is inside the cylinder x^2+y^2=8 The Attempt at a Solution my attempt was that i would turn this into polar coordinates and solve that integral but is it right? I came up with...
  26. 1

    Surface Area [Double Integral]

    I’m doing a lot of double integrals to find surface area problems, and I don’t think I’m setting them up quite right. For example, “Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 25 inside the cylinder x^2 + y^2 = 9.” I converted the sphere to a function of z: \sqrt{25...
  27. W

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    Homework Statement I believe this is intended to be a proof of the formula πrl, surface area of a cone. Homework Equations A complete volume of revolution gives you a cone - the height h is the x value on a graph, the radius r is the y value. The y intercept is zero, therefore y=r/hx ...
  28. J

    Surface area of a dome-ish roof

    Homework Statement Determine the surface area of the roof of the structure if it is formed by rotating the parabola about the axis. Homework Equations SA=\int _0^{16}{2\pi\left ( 4 \sqrt{16-y} \right ) dy} (?) The Attempt at a Solution SA=\left [ -\frac{16}{3}\pi\left ( 16-y...
  29. T

    Differentiation of surface area of a right cylinder

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  30. ElijahRockers

    How is the Formula for Surface Area of a Frustum Verified in Exercise 62?

    Ok, so I have problems understanding where they came up with the formula S=2∏rL, where r = (1/2)(r1+r2). They say the formula is verified in exercise 62. I did the exercise and showed that S=∏rL. Correct me if I'm wrong, but isn't this contradictory? I can see how they're using the...
  31. M

    Finding Surface Area Cone through integration

    Homework Statement Find the surface area of the cone with the following equations: x= u sin(a)cos(v) , y= u sin(a)sin(v), z=u cos(a) where 0<=u <=b , 0<=v<=2(pi), a is constant! The Attempt at a Solution Trying to solve this I first calculate the absolute value of the cross product of r'(u)...
  32. N

    Why is the formula for the surface area of a circle not (2*pi*r)^2?

    Actually I changed my mind and feel like it should be ((pi*r)(2*pi*r)) by my faulty thinking. Since pi*r would give you a line wrapped halfway around a sphere, I was thinking you could repeat this line in a radial pattern around the outside of a sphere (2*pi*r) times to get the surface area...
  33. W

    Minimizing Surface Area in Tin Can Design: A Mathematical Analysis

    Homework Statement Is minimiznig the area of tin used to make a can an important factor? Suppose a manufacturer wishes to enclose a fixed volume,V, using a cylindrical can. The height of the cylinder is denoted by h, and the radius of the cylinder can section by r. i)Write a function for the...
  34. P

    Formula regarding insulator thickness, and surface area covered?

    Is there a general formula to measure the amount of heat lost per second (Joules per second) given the surface area which the water covers and the width of the insulator material surrounding it?
  35. K

    Surface area dependence of radiative heat transfer

    Hi all I was just wondering about the dependence of radiative transfer from a body, on its surface area and not the volume. As per stefan's law, the variation is (StefanConstant * SurfaceArea* Temp^4) The primary source of these electromagnetic radiation is from the charges in the body...
  36. Telemachus

    Surface Area of Paraboloid Limited by Plane

    Hi there, I have to compute the surface area for V:\{ -2(x+y)\leq{}z\leq{}4-x^2-y^2 \} I have a problem on finding the surface area for the paraboloid limited by the plane. I've parametrized the plane in polar coordinates, I thought it would be easier this way, but also tried in cartesian...
  37. R

    Finding the Surface Area of a Heat Sink: A Homework Guide

    Hello I've been stuck with this for ever, can't find the relevant formulas Homework Statement Given that the surface area of the first heat sink, S1= 500 cm2 = 0.05 m2 2nd heat sink = ? The thermal resistance between p-n junction and case, RTjb = 0.6˚C/W The thermal resistance...
  38. V

    Surface area and volume of a sphere

    solve this integral The area of a circle can be found by the washer method The exact area of a washer is dA = 2 \pi r \,\,dr \,\,\,\,\,\,\,eq.1 Area of a circle is: \int 2 \pi r \,\,dr Equation 1 mulitplied by h (height) gives the exact volume of a cylindrical shell: dV = 2 \pi r...
  39. S

    Approximating surface area of hemisphere

    Homework Statement i am trying to apprixmate the surface area of a hemisphere. i am approximating by cutting the sphere into cylinders of different radius, and using their curved surface area to approximate. each cylinder will have a height of r.cos.theta and radius of r.sin.theta...
  40. A

    Multiple integral to find the contact surface area of a bolt thread

    I'm trying to determine the contact surface area of a threaded bolt so that I can make some changes to the design and maintain the same contact surface so as not to affect the performace in other ways. This presents itself as a helix, and my initial guess was that it may be best to work this...
  41. N

    Find Surface Area of Sphere Part Above xy-Plane & Within Cylinder

    Find the surface area of that portion of the sphere x^2 + y^2 + z^2 = a^2 that is above xy-plane and within the cylinder x^2 + y^2 = b^2 , 0 < b < a Solution.. i try to find fx and fy.. http://imageshack.us/f/594/33049204.jpg/" how am i going to proceed?
  42. J

    Surface Area Calculations in AutoCAD

    My problem: I calculate, using Differential Geometry, the surface area of a specific part to be 50% more than the surface area AutoCAD calculates it to be using the AREA command on an extruded solid. I am certain that my calculations are correct. I use theorems of Differential Geometry that...
  43. T

    What is the surface area of the filament? heat expansion and transfer

    Homework Statement the filament of a light bulb has a temperature of 2580Cel. and radiates 60W of power. The emissivity of the filament is 0.36. What is the surface area of the filament? Homework Equations P= \sigma*A*emissivity*T^4 A=? \sigma=\sigma\ =\ 5.670400(40)\ \times\...
  44. J

    Counter Intuitive result for Surface area of Helical Ribbon

    I am solving for the surface area of a helical ribbon that I represent as a ruled surface, the curve being the helix and the rulings being in the vertical \sigma\left(t,\varphi\right)=\left(\begin{array}{ccc} r\cos t, & r\sin t, & \omega t+\varphi\end{array}\right) I solve for the terms in...
  45. H

    Another Curved Surface Area Problem

    Homework Statement Find the area of the part of the cylinder y^2+z^2=a^2 that lies inside the cylinder x^2 +y^2 = a^2 Homework Equations The Attempt at a Solution So the first thing I did was I solved for z from the first equation to get z = Sqrt[a^2-y^2]. I took the partial...
  46. H

    What are the bounds for the double integral in this curved surface area problem?

    Homework Statement Find the area of the part of the surface z^2 = 2*x*y that lies above the xy plane and is bounded by the planes x=0, x=2 and y=0, y=1. Homework Equations The Attempt at a Solution z = Sqrt[2*x*y] Sqrt[(partial z/partial x)^2 + (partial z/partial y)^2 +1] =...
  47. A

    Calculating Surface Area on a Bounded Tetrahedron Plane

    Homework Statement 4. The domain D is a tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1 Calculate (a) The volume of the domain. (b) The x-coordinate of the centre-of-mass of the domain, assuming constant density. (c) Find, in terms of x and y the vector R from the origin...
  48. N

    Finding parametric surface area

    I was able to get an answer to this homework problem, but I have no way of verifying that it is correct. I was hoping someone more experienced than me could look over my work and let me know if I did the problem correctly. Homework Statement Find the surface area of the part of the...
  49. A

    Finding the vectorial element and surface area dS

    Homework Statement The domain D is a tetrahedron bounded by the planes x = 0, y = 0, z = 0 and x + y + z = 1 Calculate (a). The volume of the domain. [10 marks] (b). The x-coordinate of the centre-of-mass of the domain, assuming constant density. [9 marks] (c). Find, in terms of x and y the...
  50. D

    How Is the Surface Area of a Menger Sponge Calculated?

    Hi, I want to calculate the surface area of the Menger Sponge and found the following explanation online: N = number of square faces in the sponge) N[0] = 6 N[1] = 8N[0] + 4x6x1 (each face of original is split into 8, and also 4x6 for the holes) N[2] = 8N[1] + 4x6x20 (each face of N[1]...
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