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.

View More On Wikipedia.org
Recent contents Top users
  1. R

    Symbols used to represent surface area Ratio in Refrigeration

    Dear all, Does anyone know common symbols used to represent the ratio of two surface areas? The ratio is the surface area of low temp. heat exchanger divided by the total surface area of low and high temp. heat exchanger: AL/AT = area ratio, i want to find or create a symbol for. I would...
  2. S

    What is the surface area of a parabolic settling pond with a clay bottom?

    Homework Statement An industrial settling pond has a parabolic cross section described by the equation y = \frac{x^2}{80} . the pond is 40 m across and 5 m deep at the cetner. the curved bottom surface of the pond is to be covered with a layer of clay to limit seepage from the pond. determine...
  3. P

    Sphere has the minimum surface area?

    How do you prove that for a given volume, sphere has the minimum surface area?
  4. mesa

    How to derive the surface area of a sphere?

    So, I have been trying to figure out how to derive the equation for the surface area of a sphere. All attempts have resulted in colossal failure and as such are not even worth posting on the forum. I know Archimedes was the first to come up with the formula but I have not been able to find...
  5. I

    Calc II: i don't understand integral for surface area

    if we want to find the volume of a function revolved about the x-axis all we do is find the differential element.. dV=(Area)dx =\piy2 , where y=y(x) so then.. \int\piy2dx the differential element looks like this... http://imgur.com/HwrihKA,hjkKFzV,cpUPsvZ,YKIyTIB so i add a bunch...
  6. T

    Why does pressure in liquid not dependent on the surface area?

    is it because that pressure in liquid acts in all directions?
  7. R

    MHB Surface Area of an Ellipse Obtained by Rotation

    A surface is obtained by rotating around the x-axis the arc over the integral(-1,0.5) of an ellipse given by: x^2+4y^2=1 What is its surface area? Here's my solution: I use the equation: S=integral( upper bound: a lower bound: b ) 2(pi)y*[1+(f'(x))^2]^0.5 dx Since x^2+4y^2=1...
  8. J

    Optimization: maximum curved surface area of a cylinder in sphere

    θHomework Statement The attached diagram depicts a sphere with several variables: the height of the cylinder, the radius of the cylinder and an angle. All that has been given to me is the hypotenuse of a triangle used. Homework Equations To my knowledge I was told to use 2sinθcosθ=sin2θ...
  9. skate_nerd

    MHB Implicit surface area, sphere cut by two plane

    Got a little problem here, just want to make sure what I'm doing is right because it's a little different from anything I've done that is using the same formula initially. So there's a sphere denoted as \(\Omega\): $$x^2+y^2+z^2=R^2$$ and its cut by two planes \(z=a\) & \(z=b\) where...
  10. P

    Sphere surface area and radius

    I am curious about the relationship between an ever expanding sphere's radius and its surface area. how would I relate the rate of change in radius to the rate of change in the surface area.
  11. P

    Double Integration - Surface Area

    Homework Statement Find the surface area of the surface defined by: z = (2/3)(x^(3/2) + y^(3/2)), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 Homework Equations SA = ∫∫ √(fx)^2 + (fy)^2 + 1 dA The Attempt at a Solution Solved for the partial derivatives and plugged them into the formula, but got a...
  12. D

    Calculating flux through a surface area

    S is a portion of a curve with r(u,v) where 0 < u < 2 and 0 < v < 2pi I'm meant to calculate Flux of the vector field F My Calculations First found dr/du then dr/dv Using the cross product, I found N = (- u cos (v) + 5 sin (v), -5 cos (v) - u sin(v), u) Then I dot product with the given...
  13. M

    Surface Area of Revolution about x axis

    Homework Statement x^2 + (y - 2)^2 = 1 The hint given by the question was to split the function into 2Homework Equations Surface Area about x axisThe Attempt at a Solution So i did this. (y - 2)^2 = 1 - x^2 y = √(1 - x^2) +2 and y = 2 - √(1 - x^2) The range I calculated when...
  14. Petrus

    MHB Surface area of rotation about the y-axis

    Calculate the area of ​​the surface of rotation which occurs when the curve rotate in y-axe. I start with x=\sqrt{28y} then f'(x)=\frac{14}{\sqrt{28y}} so we got 2\pi\int_0^{21}\sqrt{28y}\sqrt{1+(\frac{14}{\sqrt{28y}})}^2 then I rewrite as \int_0^2\sqrt{28y}\sqrt{1+\frac{196}{28y}}...
  15. J

    Projection of surface area elements in vector calculus

    Homework Statement (i) Find the normal, n, at a general point on the surface S1 given by; x2+y2+z = 1 and z > 0. (ii) Use n to relate the size dS of the area element at a point on the surface S1 to its projection dxdy in the xy-plane. The Attempt at a Solution To...
  16. V

    Calculating Surface Area of Rotated Functions

    Homework Statement Find the surface area of the solid obtained by rotating y = √9 − X^2 , − 2 ≤ X ≤ 2 about the X-axis. Homework Equations ∏r^2 The Attempt at a Solution Y = (9 - x^2)^-1/2 Is that can do like this.. then how bout the power inside the X^2??
  17. R

    Surface area proof (vector analysis)

    edit: vector* analysis; sorry for the typo Homework Statement Given that A = ||ru||2, B = ru\bulletrv, C = ||rv||2 surface area of S is Area(S) = \int^{d}_{c}\int^{b}_{a} = sqrt (AC - B2) dudv The Dirichlet energy can be thought of as a function as follows E(S) =...
  18. T

    Given a surface defined in parametric equations, find the surface area

    Homework Statement The Attempt at a Solution So I am trying to find the gradient because in class we were taught that the surface area is the magnitude of the gradient divided by the magnitude of the dot product between the gradient and normal to the projected surface. I noticed...
  19. A

    Surface Area of a Sphere in Spherical Coordinates; Concentric Rings

    Hey, folks. I'm trying to derive the surface area of a sphere using only spherical coordinates—that is, starting from spherical coordinates and ending in spherical coordinates; I don't want to convert Cartesian coordinates to spherical ones or any such thing, I want to work geometrically...
  20. S

    Surface area of a curve

    Homework Statement The given curve is rotated about the y-axis. Find the area of the resulting surface. y = 3 - x^2 0 \leq x \leq 4 Homework Equations A_{y} =2\pi\int_a^b x \sqrt{1+\left(\frac{dx}{dy}\right)^2} dy The Attempt at a Solution now we need to write x in terms...
  21. C

    Surface Area of a Sphere without double integral

    Is it possible to come up with a derivation of the surface area of a sphere without using a double integral? Most of the ones I've found seem to involve double integrals; For example, this was given as the "simplest" explanation in a thread from 2005: S=\iint...
  22. P

    Need help with finding surface area of a solid of revolution

    1. Use the integration capabilities of a graphing utility to approximate the surface area of the solid of revolution. (Round your answer to four decimal places.) Function: y = sin(x) Interval: [0, pi/4] revolved about the x-axis 2. Use the area of a surface of revolution...
  23. W

    Pappus' Theorem (surface area)

    Homework Statement Use Pappus' Theorem for surface area and the fact that the surface area of a sphere of radius c is 4(pi)c^2 to find the centroid of the semicircle x= sqrt ( c^2 - y^2) Homework Equations S = 2 (pi) * p * L where s=surface area; p=distance from axis of revolution...
  24. K

    Exploring Flying Objects: Volume & Surface Area

    Homework Statement Suppose we consider different flying objects, and that each object is characterized by a linear dimension l. Part A: Use dimensional arguments to show that the volume V scales with size as V \sim l^{3} and that the surface area scales as S \sim l^{2}. Part B: Show...
  25. E

    Correlation between friction force and surface area

    As far as I know, friction force is equal to the product of the normal force and coefficient of friction, hence is independent of surface area. So why is it that race cars have wider tyres than conventional vehicles?
  26. S

    Difference Between Surface Integrals and Surface area using double integrals .

    Hi all, Thanks for response :) I Dont really understand what is surface integrals ?? and its difference with Surface Area using double integrals. Can anyone help ? thanks a lot...
  27. S

    How friction depends on surface area

    I'd read that friction is independent of the surface area of the bodies in contact. But somewhere in the internet I found that this explanation was just a good approximation and that friction actually depends on area. Can anyone explain a bit more on this?
  28. S

    Why Can't We Calculate Sphere Surface Area with Infinitesimal Cylinders?

    We know that we calculate the volume of sphere by taking infinitesimally small cylinders. ∫ ∏x^2dh Limits are from R→0 x is the radius of any randomly chosen circle dh is the height of the cylindrical volume. x^2 + h^2 = R^2 So we will get 4/3∏R^3 Now the question is why cannot we...
  29. R

    Help with question regarding surface area

    Homework Statement Oil is released from a submerged source at a constant flow of rate of Q=(0.1 m3)/s. Density of oil (p=870 kg/m3) is less than that of water and since oil is only sparingly soluble in water a slick will form on the water surface. Once its formed it will have a tendency to...
  30. U

    Finding surface area with a function. Double Integration.

    Homework Statement Calculate ∫∫f(x,y,z)DS for the given surface and function. Part of the plain x+y+z=0, contained in the cylinder x^2+y^2=1 f(x,y,z)=z^2. Homework Equations ∫∫F(x,y,z)Ds=∫∫F(g(u,v)*||n(u,v)|| N= TuXTv Tu= G(u,v)(du); Tv is the same only the derivative is with respect to v...
  31. L

    Finding the surface area of a curved object using calculus

    1. Homework Statement I need some help with a surface area of a solid. The solid is made from rotating the line y=x^2 around the x axis. So it's sort of like a cone or a horn. Here are my steps: 2. Homework Equations Surface of revolution formula Integrate 2∏r times the square root of 1...
  32. M

    Question about calculating surface area

    So if we are looking to find the surface area of a solid of revolution formed by rotating a curve about a line, we can use the following: S = ∫2(pi)yds (if rotating about the x-axis) FOr example, say our curve is y=x2 and we want to find the surface area of the solid of revolution...
  33. V

    Surface area of solid of revolution (no calculus)

    Homework Statement Consider the region of the x y plane given by the inequality: x^2 + 4x + y^2 - 4x - 8 ≤ 0; If this region rotates an angle of π/6 radians around the line given by the equation x + y = 0, it will create a solid of revolution with surface area equal to (i)...
  34. L

    Surface area of a square and a tube

    Ok, so if I have a square that is exactly 10 inches by 10 inches, then the surface area is 100 square inches exactly. But if I roll up that square into a tube and calculate its surface area, it's 2∏r times the length. And since the calculation involves ∏, that means I won't get an exact answer...
  35. S

    Perimeter and surface area of skewed end semi cylinder

    I want to calculate the surface area of semi cylinder with one or both ends skew at a given angle. To calculate the area/perimeter i need to find out the end profile of the semi cylinder. If the skew plane is perpendicular to the cylinder axis, the cut surface would be half ellipse with major...
  36. soothsayer

    Surface area of a spherical cap

    Homework Statement Calculate the area of a circle of radius r (distance from center to circumference) in the two-dimensional geometry that is the surface of a sphere of radius a. Show that this reduces to πr2 when r << a Homework Equations Surface area of a spherical cap = 2πah = π(r2 +...
  37. L

    Theoretical situation - surface area vs. friction

    You are pushing a wooden crate across the floor at a constant speed. You decide to turn the crate on end, reducing by half the surface area in contact with the floor. In new orientation, to push the same crate across the same floor with the same speed, the force that you apply must be about: A)...
  38. A

    MHB Finding surface area with volume

    Ok, so I did a test today for advance functions and there was a question: the volume of a pop can is 350ml, find the minimum surface area and determine the dimensions. Where I got stuck:350=pi(r)^2*h h=350/pi(r)^2SA= 2pi(r)^2+2pi(r)( h) SA=...
  39. A

    Finding surface area with volume

    Ok, so I did a test today for advance functions and there was a question: the volume of a pop can is 350ml, find the minimum surface area and determine the dimensions. Where I got stuck: 350=pi(r)^2*h h=350/pi(r)^2 SA= 2pi(r)^2+2pi(r)(h) SA= 2pi(r)^2+2pi(r)(350/pi(r)^2)...
  40. D

    Silicon Dioxide surface area

    Homework Statement Grains of fine California beach sand are approximately spheres with an average radius of 50 μm and are made of silicon dioxide, which has a density of 2.6 × 103 kg/m3. What mass of sand grains would have a total surface area (the total area of all the individual spheres)...
  41. S

    Calculating Surface Area of a Torus

    So the first question is to find the surface area of a torus generated by rotating the circle (or shall I say semi circle) y=√r-x2 around y=r if the idea is to find the surface are of the half torus and then multiply by 2, wouldn't it be the same for the circle (or shall I say semi circle...
  42. R

    Surface area and line integral

    I met a proof problem that is as follows. ##\bf a = ∫_S d \bf a##, where S is the surface and ##\bf a ##is the vector area of it. Please proof that ##\bf a = \frac{1}2\oint \! \bf r \times d\bf l##, where integration is around the boundary line. Any help would be very appreciated!
  43. S

    What Is the Correct Surface Area of a Curve Rotated Around the X-axis?

    Homework Statement Find the area of the surface obtained by rotating the curve y=x3, 0≤x≤2 about the x-axis. Homework Equations \begin{equation*} SA = \int_{0}^{2} 2 \pi y L \end{equation*} The Attempt at a Solution SORRY, I don't know how to use LaTeX yet. ∫2∏y√(1+(dy/dx)2)dx from 0->2...
  44. D

    Why is the Height of Each Cylinder in a Cone dS and not dH?

    Hello, I have actually asked a similar question before, but I just realized something and I want to edit the question now: I am trying to derive the formula for the lateral surface area of a cone by cutting the cone into disks with infinitesimal height, and then adding up the lateral areas...
  45. D

    Deriving the volume and surface area of a cone

    Hello, this is my first time posting on physics forums, so if I do something wrong, please bear with me :) I am trying to derive the formula for the lateral surface area of a cone by cutting the cone into disks with differential height, and then adding up the lateral areas of all of the...
  46. N

    Surface area of a solid of revolution

    Homework Statement Having recently learned the disk/shell/washer method for finding the volume of a solid of revolution, I'm trying to apply similar methods to derive the formula for the surface area of a cone (and hopefully after that, that of a sphere). The region that is revolved around...
  47. C

    Integral to find the Surface area

    For a given function f(x)=x, if we rotate this function from [0,1] around the x-axis, we'll have a cone... why can't I find its surface area by adding the perimeter of all the circunferences from [0,1] with radius = f(x)?Something like 2*pi∫xdx? The formulas says we have to do 2*pi∫x*√(i+y'²)dx ...
  48. J

    Dimensions - Disc Brake Surface Area

    Dimensions -- Disc Brake Surface Area During frequent braking under race conditions the disk brake rotors on the car described above reach a temperature of 500C. These disk brakes rely on forced convection to cool them. The dimensions of each disk rotor are: outer radius 130mm; inner radius...
  49. W

    How the rate of heterogeneous reaction changes with the surface area of catalyst

    how the rate of heterogeneous reaction changes with the surface area of catalyst. can anyone tell me about the equation relating both quantities?
  50. M

    Surface Area of a Multivariable function

    I don't know how to calculate the surface area after setting everything up. I have tried both MAPLE 15 program and wolfram alpha, but I can't find the answer. I have the function: f(x,y)= x*(y^2)*e^-((x^2+y^2)/4) The form I found for surface are was the square root of (sum of squares of...
Back
Top