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. D

    Calculating Surface Area of a Revolution Rotated About the Y-Axis

    (my first dealings with latex.. so bare with me if this looks a little messed up at first :rolleyes: ) Homework Statement Find the surface area for the equation: x = 3y^{4/3} - \frac{3}{32}y^{2/3} with bounds -216 \leq y \leq 216 rotated about the Y-axis. Homework Equations \int^a_b 2\pi...
  2. S

    Proving Surface Area Formula with Multiple Integrals

    Surface Area (help me to prove something:) I was studying a bit about multiple integrals and found this theorem: If we have function z=f(x,y) which is defined over the region R, surface S over the region is S=\iint_R\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial...
  3. A

    Rate of increase of the surface area

    a spherical balloon is being inflated. find the rate of increase of the surface area (S=4 pie r squared) with respect to radius r when r is (A) 1 ft, (B) 2 ft, (C) 3ft. Here's what i did i found the derivative of s and i put 4 2R and then i plugged in the numbers in R and i got 8 ft...
  4. A

    -friction Is Independant Of Surface Area-

    ----friction Is Independant Of Surface Area------ >> can anyone tel y friction is independant of surface area/length ... thx an regards, arun
  5. B

    Surface Area Calculation: u^2+v^2≤1

    Could someone help with the following? I am asked to find the surface area of the following surface with parametric equations x = uv, y= u+v, z = u-v, and u^2+v^2≤1. So d/du is <v,1,-1> and d/dv is <u,1,-1> And the cross product is -2i + (u+v)j + (v-u)k. So the magnitude of the vector...
  6. B

    Solve Surface Area Problem: Sphere & Cylinder

    I am wondering if someone could help me with the following? I am supposed to find the surface area of the part of the sphere x^2 +y^2+z^2 that lies inside the cylinder x^2+y^2 = ax. If I wanted to write a parametric equation for the sphere, I would use x = ρsinφcosθ and y = ρsinφcosθ and z...
  7. S

    Surface area and volume of a cylinder

    Could a tutor please check my work? question: What is the surface area and volume of a pressure vessel in the form of a cylinder with each end in the form of a hemisphere, if the overall length is 12 meters and the diameter is 3 meters. solution: given: radius = 1.5m diameter = 3m...
  8. G

    Calculating Surface Area of Revolution for a Rotated Curve

    Homework Statement Find the area of the surface obtained by rotating the curve about the x-axis: y=cos 2x, 0<=x<=pi/6 Homework Equations Surface area about the x-axis = Integral of 2pi * f(x) * sqrt(1+[f'(x)]^2) dx The Attempt at a Solution I think I set up my integral correctly, so...
  9. B

    Surface Area Problem: Find Area of Paraboloid Cut by Plane y=25

    Homework Statement I am wondering if someone could help me with the following? I am supposed to find the area of the finite part of the paraboloid y = x^2+z^2 that's cut off by the plane y = 25. Now, wouldn't this be the same as the paraboloid z = x^2+y^2 that's cut off by the plane z = 25...
  10. B

    Best dimensions for maximum surface area

    I was doing some math problems involving surface area and maximum dimensions and then I wondered: Suppose you are given the surface area of a rectangular box but none of its dimensions. Is it possible to find the best dimensions (x,y,z) that would give the maximum volume of the box? I was...
  11. P

    Proving Sphere has Minimal Surface Area for Volume

    Could someone give a clue to how I could prove the sphere has minimal surface area for a given volume? Note this is not a homework problem. I saw in a chemistry textbook that water droplets tend to be spherical because each water molecule has a force directed inwards. In order to minimise the...
  12. C

    Most Accurate Estimate of Surface Area of a Bottle?

    What is the most accurate way to estimate the surface area of a cylindrical bottle that decreases in diameter from its wides point (the body) to the narrowest point ( the neck and cap)? It seems like we did similar problems in calculus, however: 1. I don't remember any of it, and 2. I think...
  13. S

    How do I integrate surface area using rectangular or polar coordinates?

    Just had my test on Vector Fields and there was one question which really confused me. It asked to find the surface area of the parabaloid z = 9-x^2 -y^2 which is above the cone z = 8Sqrt[x^2 + y^2]. My memory told me to use the differential in rectangular coordinates and then convert to...
  14. G

    Surface area of a rotated curve

    Here's the question: Find the area of the surface obtained by rotating the curve http://ada.math.uga.edu/webwork2_files/tmp/equations/18/d733a6e52ad8ca260230969bdc3f401.png from x=0 to x=9 about the x-axis. I'm supposed to parametrize the curve, using rcos(theta) as x and...
  15. S

    Surface Area of a Sphere

    I've been looking at this method here: http://planetmath.org/encyclopedia/6668.html I was wondering at the last step before the "Note on multi-valuedness" if you wanted to obtain the general formula for the surface area of a sphere 4 \times \pi \times r^2 with a radius of well r what...
  16. B

    Surface area of the node during Electrolysis

    While doing electrolysis, Will the surface area of the node effect how much hydrogen is produced? Will it effect how well the current can pass through the water? Thank you
  17. J

    Surface Area in Polar Coordinates

    Find the surface area of the surface z=cosh(sqrt(x^2+y^2)) above the region in the xy plane given in polar coordinates: r is between 0 and theta theta is between 2 and 4 Ok. I used the formula: Surface area equals the square root of the partial derivative of x squared plus the partial...
  18. S

    Surface Area of a Cylinder

    May I have a tutor check over my question and solution? Problem Two rectangular sheets 20cm x 24cm and 15cm x 30cm are to be rolled to form cylinders. What is the height and diameter of the cylinder with maximum surface area that can be formed using either of these sheets? My Solution...
  19. J

    Calculating Surface Area of a Cone with a Vertex in the XY-Plane

    Find the surface area of the cone z=2sqrt(x^2+y^2) and above a region in the xy-plane with area 5. If anyone could help me with this problem, I would really appreciate it. Thanks. Mike
  20. E

    Surface area of a cube with the length of each edge equal to x+1

    I am having trouble finding the surface area of a cube with the length of each edge equal to x+1. can some one help?
  21. K

    Surface w/ max volume and min surface area

    I'm attempting to solving this problem but do not know how to begin. Any help would be appreciate. What geometric surface encloses the maximum volume with the minimum surface area? How would you prove it?
  22. U

    Finding Surface Area of Sphere Above Cone

    Surface area integral sorry, this is not about flux integration... but surface area! sorry about the title! Find the surface area of the part of the sphere x^2+y^2+z^2=36 that lies above the cone z=\sqrt{x^2+y^2} z=\sqrt{36-x^2-y^2} A(S)=\int\int_D \sqrt{1+\left( \frac{\partial...
  23. B

    Finding surface area of a sphere

    Can someone tell me how to find the surface area of a sphere when given the Area? :confused: thanks
  24. G01

    Surface Area of Revolved Curve: An Intriguing Challenge

    Find the surface area when this curve is revolved around the x ax x = 1/3(y^2 + 2)^3/2 [1,2] I set it up both ways and i get two really complicated integrals. 2 \pi/3 \int_1^2 \sqrt{\frac{y^2+y^4}{(y^2+2)^{2/3}}} dy Yeah I can't figure out how to do this integral, and I am thinking there...
  25. T

    Minimum Surface Area Cylinder using Euler-Lagrange Equation?

    So, I've been reading Thornton and Marion's "Classical Dynamics of Particles and Systems" and have gotten to the chapter on the calculus of variations. In trying the end of chapter problems, I find I'm totally baffled by 6-9: given the volume of a cylinder, find the ratio of the height to the...
  26. J

    Surface Area and Volume Relationships?

    So, I had a question for all of you, regarding the relationships between area and circumference, and surface area and volume. For the longest time I was confused as to how these two quantities were related. I saw that the derivative of the area of a circle was equal to its volume, but then...
  27. S

    So I had a problem asking to find the surface area of a sphere with radius=a

    So the first thought that occurred to me was to use a surface integral I got 2pi*a^2, which is half the SA, if I used the surface integral is it possible or likely that I just found half the SA and can then multiply by 2? Or something. Actually, I think I see what I did, for my sec(gamma) I...
  28. T

    Finding Surface area of a Parametric Curve

    Can someone please help me with this question? x = 1-sint, y = 2+cost, rotate about y = 2 Find the surface area of the parametric curve. I don't know how to do it with y=2, I only know how if the question askes for rotating about the x-axis. The answer to the question is 2(pi)^2.
  29. J

    Calculus of parametric equations (finding surface area)

    I was wondering what the surface area would be when the curve: x=e^tsint, and y=e^tcost where (t) is greater than or equal to (0) and (t) is less or equal to pi divided by (2). when it is revolved about a) the x-axis b) the y-axis (approximation...
  30. D

    Surface area of a sphere problem

    Sphere 1 has surface area A1 and volume V1, and shere 2 has surface area A2 and volume V2. If the radius of shere 2 is four times the radius of shere 1, what is the ratio A2/A1 of the areas?
  31. I

    Where is the Angle Theta Measured in a Cylinder's Surface Area?

    I want to find the surface area of only a part of the cylinder.what i mean is that the surface area that is under an angle theta in the cylinder.
  32. Y

    Calculating Surface Area around a Vertical Line: Tips and Tricks"

    Surface area help please... How do you find a surface area of a function around a vertical line? For example: surface of \ y=x^3 between \ 0<= x <= 3, rotating around \ x=4? I tried the formula for finding surface area, but I confuse with that vertical line x=4... what should I do?
  33. B

    Calculating Surface Area & Volume of 3D Function

    Area/Volume Question Given the function on a 3D coordinate system: y = - z\sin \left( {xz} \right) where \left| {x - \frac{\pi } {2}} \right| \leqslant \frac{\pi }{z},\;z \ne 0 How do find the surface area of the figure integrated from z=a to z=b (where 'a' and 'b' are constants) ...
  34. V

    Why Don't Surface Area & Volume Ratios of Spheres Match?

    Howcome the ratio of the surface area of spheres are not equal to the ratio of volume of spheres? For example if I had two spheres, with an aspect ratio of 1:4 when comparing surface areas, yet I have 1:8 between those same spheres when comparing volume. This doesn't make sense to me...
  35. D

    Surface Area (Surface of Revolution) - Discrepancy

    So this question has been bothering me for a very long time... but only recently have I mustered up courage to register to ask it. Before though, let me draw a parallel analogy, so you can see where I'm coming from. When you take the volume of a region enclosed by one or more functions...
  36. Cyrus

    Calculating Surface Area Using Double Integrals

    Hi, I have a question on the method of calculation of the surface area of a surface. I am using "Calculus Concepts and contexts by stewart", chapter 12.6. In it, he goes on to explain how to calculate the suface area of a surface as a double integral by using approximations. He breaks up...
  37. DaveC426913

    Unravelling Surface Area of a Dome

    Trying to figure out the answer to another thread. What is formula for the surface area of a dome? Googling got me 2\pi r h (where h is the height of the dome above its slice through the sphere). Is that right? Ultimately, I'm trying to figure out how the area changes as a function of...
  38. DaveC426913

    What is formula for the surface area of a dome?

    Trying to figure out the answer to another thread. What is formula for the surface area of a dome? Googling got me 2\pi r^2 h
  39. G

    Surface area of atomized molecules?

    I have the following problem, which seems easy, I just cannot get my brain around it... Assume that 30.0 cm^3 of gasoline is atomized into N spherical droplets, each with a radius of 2.00\times 10^{-5} m. What is the total surface area of these N spherical droplets? How do I find the...
  40. M

    Finding Surface area using Newton's Law

    As part of a homework assignment, I am required to research Newton's Law of Cooling and evaluate it with three practical examples. One must be time of death. So far i have created a time of death example, and used an example to cool food. However the last one i heard someone say is that you...
  41. W

    Why Dropping the Sqrt{1 + 1/x^4} dx Matters for Surface Area

    ok I know how to solve for the volume no prob, but when you find the surface area, in all the proof they always drop the root, because something about being bigger than one Sqrt{1 + 1/x^4} dx can someone explain why they drop that, and how it's possible to have finite volume and infinite...
  42. L

    Calculating Total Surface Area of a Pool with Varying Depths

    G'day all I'm new here this is my first post. I just found this forum in hope that you all can help me and I've hopfully come to the right place. i'm pretty bad at maths (actually shocking at it) and this may be a dumb question but any help would be appreciated. I'm doing total surface area...
  43. K

    Help Finding Surface area with Double Integrals. (work shown)

    find the surface area of f(x,y)=\sqrt{x^2+y^2} above the region R=\{(x,y):0\leq f(x,y) \leq 1\} well.. here's what the answer should be.. \sqrt{2} \ \theta 1. formula for Surface area: \int_{R} \int \sqrt{1+f_{x}(x,y)^2+f_{y}(x,y)^2} \ dA 2. next i need to find the region: so if...
  44. M

    Surface Area of y=(x)^1/2 Rotating About the x Axis

    i've been stuck on this problem for an hour now. how do you find the surface area for y=(x)^1/2 when 0_<x_<2, rotating about the x axis?
  45. holly

    Surface Area vs Volume in Biology

    Can someone pull a rabbit out of a hat for me & explain why just because an object (like a cell or an animal) gets bigger, its surface area doesn't get proportionally bigger? I made myself a little chart where I took a sphere & found the vol & the s.a. at 1 cm radius, at 2 cm radius, and 3 cm...
  46. tandoorichicken

    Parametric Equation for Surface Area of Revolution

    I know that the equation for the surface area of any solid of revolution around, say, the x-axis is SA = 2\pi\int_{a}^{b} y\sqrt{1 + (\frac{\,dy}{\,dx})^2} \,dx What I need is the same formula except in parametric terms, like if the problem was given in terms of x(t) and y(t). Any takers?
  47. P

    Find the Surface Area of a Sector of a Sphere: Trig Equations

    I'm looking for a long list of trig equations. Mainly as of right now I'm looking for the equation to find the surface area of a sector of a sphere? Any help would be great, Thanks Philip
  48. I

    Surface area and total area

    What's the difference b/w surface area and total area? For example, the surface area of a cylinder is A = 2 pi rh while the total area of a cylinder is A = 2 pi rh x 2 pi r^2. Thanks for the help =)
  49. D

    Resistance: Non-constant Surface Area

    I think I'm on the verge of a breakthrough on this problem, but it's just not coming. Please tell me where my approach goes wrong or whether I'm correct but don't know how to integrate the resulting equation properly. The problem reads: A circular disk of radius r and thickness d is made...
  50. A

    Square-based pyramids surface area

    Among all square-based pyramids which have volume V = 1, which one has the smallest surface area?
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