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. K

    Surface area of a part of z=xy in a cylinder?

    Homework Statement What is the surface area of the part of z=xy that lies in the cylinder given by x^2+y^2=1? Homework Equations SA=(double integral) sqrt(Fx^2+Fy^2+1)dA The Attempt at a Solution I've graphed it but I don't know what to integrate or the bounds.. :/ any help? thank you!
  2. S

    Surface area of a sphere using integrals

    Hello dear colleagues! Yesterday i was trying to proof the surface area of a sphere formula, then i got some problems. I know that something is seriously wrong in this concept, but i can't tell what exactly is wrong. Could you guys help me please? I just thougt about a hollow sphere, then we...
  3. C

    Does Wrapping a Whip More Times Increase Its Ability to Support Weight?

    Currently I'm doing a physics project based on Indiana Jones. I was curious if the whip, when wrapped around a tree branch, can support a human weight. However I realize, depending on how many times the whip was wrapped around the branch, the friction force will increase. Yet based on the...
  4. N

    Pappus Theorem surface area

    determine the volume and surface area obtained by rotating the area i know how to obtain the volume but i am confused on how to obtain the area, i have the solutions but i want to learn how to do it i know that the area of the rectangle is 1500 and the semicircle is -353.42 so the total...
  5. A

    Surface area of a cylinder question

    Homework Statement This is actually for my Physics 211 class but the question is very simple. We were asked to construct a cylindrical space station to specific parameters which are not important to the question. I know that my cylinder has a radius of 399.97m and a rotational period of...
  6. F

    Surface Area obtained by rotating a curve?

    1. Let the area of the surface obtained by rotating the curve x = 3e^(2y) from y=0 to y=1 about the y-axis 2. S= \int_{y_1} ^{y_2} 2 \pi x ds \ where \ ds=\sqrt{1+ \left( \frac{dx}{dy} \right) ^2} dy 3. S=2\pi \int _{0} ^{1} x\sqrt{1+\left(\frac{dx}{dy}\right)^2} dy Let u=6e^{2y}...
  7. K

    Surface Area of a Solid of Revolution

    Homework Statement Find the area of the surface generated when you rotate the parabola y=x2 0 less than or equal to x less than or equal to the square root of k, around the y-axis. You should end up with a simple formula in terms of the constant k. Homework Equations S=2\pi\intyds...
  8. X

    Understanding the concept of the formula for surface area

    Hey all, I was reviewing the concepts of how to derive the formula for the surface area, namely.. SA=\int \int \left\|\frac{\partial \widetilde{r}}{\partial u} \times \frac{\partial \widetilde{r}}{\partial v}\right\| dA (I didn't know how to make vector arrows so I used the tilde sign.)...
  9. G

    Solving spherical surface area

    For r = 3, 0 < theta < pi/2, 0 < phi < pi/3, find the surface area of the sphere. The answer given is 9pi, but I can't seem to work it out. Below is my working: 2\int_{r=0}^{3}\int_{\phi=0}^{\frac{\pi}{3}}r\sin\theta dr d \phi + 2\int_{r=0}^{3}\int_{\theta=0}^{\frac{\pi}{2}}r dr d \theta +...
  10. G

    Calculating Surface Area of Sphere for r = 3, 0 < theta < pi/2, 0 < phi < pi/3

    For r = 3, 0 < theta < pi/2, 0 < phi < pi/3 \int_{\theta=0}^{\frac{\pi}{2}}\int_{\phi=0}^{\frac{\pi}{3}}r^2sin\theta d \theta d \phi =R^2 [-cos\theta]_{0}^{\frac{\pi}{2}} [\phi]_{0}^{\frac{\pi}{3}} =(3^2)(1)(\frac{\pi}{3}) = 3\pi The above is my working. The answer should be the same as...
  11. MathWarrior

    Soild Revolutions, surface area etc.

    In calculus when you learn solids of revolution in relation to surface area or volume is there any specific uses that someone could enlighten me on in which this would be used. The only example I can think of is a lathe, and calculating its mass using the volume. Other then that I haven't seen...
  12. K

    Calculate the surface area of 1 g of TiO2 powder

    Homework Statement Calculate the surface area of 1g of powder with a size of 100micrometer. Assume the particles are spherical. Then estimate the percentage of TiO2 molecules at the surface of the particle (relative to the total number of TiO2 molecules in the volume of the particle) for the...
  13. M

    Measuring surface area between points

    Homework Statement I want to calculate the surface area, mapped with points. For starters, I've mapped the edge of a circle with 200 points, then created a few more circles, each with 200 points. The problem is I get different surface areas for the same size circles. The only difference...
  14. B

    Finding Surface Area of a Solid of Revolution

    Homework Statement Find the surface area generated by rotating y=5-4x^(3/2), 0\leq x\leq 1 about x=2.Homework Equations SA = 2\pi\int_{a}^{b}(r\cdot ds)dx The Attempt at a Solution I simply filled in the formula for the given question, and I'm getting stuck at integration time. SA =...
  15. V

    Parametrize and find the surface area

    Homework Statement a)parametrize the upper surface of that portion of the sphere x^2 + y^2 +z^2 = 4 contained within the cylinder, x^2 + y^2 = 2y b)find the area of the surface Homework Equations how to find the range for phi. for paramaetrizing into spherical coordinates The...
  16. S

    Surface Area and Friction in Static Friction?

    Homework Statement How does the surface area of an object affect the force of static friction? I'm trying to figure out whether two different objects with equal mass and different surface areas requires the same amount of applied force or different amounts. Homework Equations I know of...
  17. W

    Maximize Your Engine's Power with Improved Air Intake Surface Area

    I have been thinking about the large number of "cold air intakes" and why they are such an overwhelming trend. I question their validity for a couple of reasons. First, the stock airbox on all new cars brings cold air in, usually from the fender, and is sealed off from the engine bay, usually...
  18. S

    Surface area of solid of revolution with undefined derivative

    Hello there. Suppose I have a function: y=3x^{2/3}-1 I want to find the surface area of the solid formed when the part of the curve between x=0 and x=8 is revolved about the x-axis.The curve crosses the x-axis at a point (1/3)^{3/2} The derivative of the function is...
  19. A

    Setting up an integral for surface area

    Homework Statement Set up an integral that represents the area of the surface generated when the region is bounded by x + y2 = 1 and the y-axis, then rotated about the y-axis. All in one-variable. Homework Equations SA = 2pi \int xds possible x2 + y2 = r2 The Attempt at a Solution...
  20. J

    Surface area of the boundary enclosed by surfaces

    Homework Statement Find the area of the surface that is the boundary of the region enclosed by the surfaces x^{2}+y^{2}=9 and y+z=5 and z=0 Homework Equations A(S)=\int\int_{D}\left|r_{u}\times r_{v}\right| \; dA The Attempt at a Solution I am really confused as to what he...
  21. C

    Optimization word problem - minimizing surface area to find least expensive tank

    Homework Statement A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank covers $10 per square meter for the base and $5 per square meter for the sides, what...
  22. J

    Surface area and pressure relation to friction and wear

    not sure if my title is the most accurate but here is what I am asking. I will give general dimensions and weights in case it makes any difference. **EDIT** This is a pnuematic gun if you couldn't tell and if it makes any difference.** You have a hollow aluminum tube ~7/8" I.D. Inside the tube...
  23. F

    Surface area in spherical co-ordinates

    I have a generalised function in spherical polar co-ordinates that describes the radiant intensity in any given direction of a point source emitter. I need to calculate the fraction of the total radiation that is emitted into a region of space specified by some min to max range of elevation...
  24. Telemachus

    How Do You Calculate the Surface Area of x^2-y^2-z^2=0 with Given Constraints?

    [Urgent] Surface area Homework Statement Hi there, sorry for the hurry, I had a big problem this weekend and I couldn't study. I have my exam tomorrow, and need help with this. I'm finishing with the topic, and I have almost studied all what I had to study, but this exercise resulted...
  25. B

    Calculating Total Height of Cylinder with Two Identical Cones on Each End

    a cylinder has two identical cones one each end. the volume of the object is 50(pie/3) find the total height. so 2h+k=height. where to from there
  26. A

    Maximizing Liquid Surface Area: No Work Required

    I want to know how to increase the surface area of a liquid without doing any work. Heating would be counted in work done.
  27. Shackleford

    Minimize surface area - calculus of variations

    I can't get rearrange the last equation into a nice form to integrate with respect to x to minimize the surface area. http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-17122502.jpg?t=1287336976...
  28. J

    How can the current density and magnetic field affect Earth's total current?

    Homework Statement Near Earth, the density of protons in the solar wind (a stream of particles from the Sun) is 8.60 cm-3 and their speed is about 490 km/s. (a) Find the current density of these protons. 6.74E-7 A/m^2 (b) If Earth's magnetic field did not deflect the protons, what total...
  29. S

    Calculating Surface Area to Volume Ratio for Emulsified Droplets

    Homework Statement Suppose a lipid droplet is essentially a sphere with a diameter of 1 cm. For A – D below, the area of a sphere is 4π r2, and volume is 4/3πr3. For π, use a value of 3.14. Show your calculations neatly in the space provided to receive credit for your answer. Round all...
  30. L

    Volume of a cube as a function of surface area

    Homework Statement Express the volume of a cube as a fuction of the total surface area Homework Equations Surface area = 6x^2 Volume = x^3 The Attempt at a Solution So, we need to manipulate the surface area equation so we can use it to find volume right? SA = 6x^2...
  31. O

    Maximum Surface Area of a Flat Surface

    What is the pattern that maximizes the ratio of surface area of a flat surface to the volume of material used in this surface? The surface has to be flat on a macroscopic scale, but on the microscopic scale could have dimples or other imperfections that increase the SA/Volume ratio. I am...
  32. H

    Volume and surface area of the sphere using integration ?

    volume and surface area of the sphere using integration ?? i was trying to find the surface area of the sphere using integration, ( by revolving circle on the x-axis ) the thing is it doesn't work as the volume problem. i mean in volume problem to get the volume of the sphere, you would...
  33. H

    Volume and surface area of the sphere using integration

    Homework Statement i was trying to find the surface area of the sphere using integration, ( by revolving circle on the x-axis ) the thing is it doesn't work as the volume problem. i mean in volume problem to get the volume of the sphere, you would start with circle and start slice it...
  34. I

    Surface Area of a Sphere by Integration

    Homework Statement Say the sphere of radius "a" is made out of various rings with height R(x) and thickness dx. Adding up all of the rings will form a sphere, and in order to do that, I have to integrate. Homework Equations Trigonometric Substitution: \frac{x}{a}=sin \theta dx=a cos...
  35. G

    Surface area of a open box

    Homework Statement Ok this is from a summer assignment that I had for calculus yet it's not calculus but it was graded by my calculus teacher so maybe there's something she knows that I don't from a calculus standpoint that allowed her to mark me wrong for this so here we go... Write a...
  36. J

    Relating Surface area to volume

    How do I express the surface area of for instance a sphere in terms of volume?
  37. U

    Surface area to volume ratio problem

    Homework Statement Find the surface area of a sphere whose volume is .2 m^3. For full credit, do it without finding the radius. Homework Equations This can be solved using v= 4/3(pi)r^3 and sa= 4(pi)r^2, but my prof doesn't want that. The Attempt at a Solution so if you...
  38. M

    Surface Area of a Sphere without Caluclus Question

    Hi, I was just trying to find the surface area of a sphere without calculus, just for fun, but I ran into a little problem. The way I wanted to do it was by imagining a tennis ball, and cutting it in half forming a hemisphere. Then I imagined flattening out this hemisphere would preserve the...
  39. T

    Calculus II: Surface area revolving around an axis

    1. Find the area of the surface generated by revolving y = (sqrt)(2x-x^2), x = [0,1] a. About the x axis b. About the y-axis This problem is from the section about surface area revolving around an axis. I actually did the problem but I am not sure on my answer. For around the...
  40. N

    Surface area of an event horizon and irreducible mass

    Homework Statement Show that a black hole's event horizon will never shrink due to a Penrose process. Homework Equations M_{ir}^2 =\frac{1}{2}(M^2 + \sqrt{M^4-J^2}) The Attempt at a Solution It is easy to show that the irreducible mass will never decrease, and I know of a result that...
  41. J

    Surface area of basket ball full of sand

    Homework Statement A regulation NCAA basketball is a sphere of radius approximately 12 cm. Grains of Texas beach sand are approximately spheres of radius 25 μm. If the inside volume of a basketball were completely filled with sand (assume perfect packing with no empty space) what is the...
  42. S

    Geometric understanding of integration / surface area of sphere

    Hi everyone, I've browsed around the forum a bit and found that others have had the same problem as me, however, none of the answers help me a lot, so I thought to post a more specific question, I hope you don't mind. I'm having a problem with the surface area of a sphere, probably because...
  43. R

    Surface Area of a part of a plane inside an ellipsoid.

    Homework Statement Find the surface area of that part of the plane 9x+10y+z=6 that lies inside the elliptic cylinder \frac{x^2}{25} +\frac{y^2}{100} =1 2. The attempt at a solution Once again I was just told that the surface area would be equal to the double integral of the area...
  44. S

    Relationship between a given volume for a cylinder and the minimum surface area.

    I currently have a question that i am struggling with it is: Propose a mathematical model in the form of an equation desribing, in general terms, the relationship between a given volume for a cylindrical container and the minimum surface area of material required to make it.. i am...
  45. N

    Water displaced against surface area

    If 2 differently shaped objects are submerged in water, say one is rectangular and other spherical, and both displaces the same volume of water, would the surface area of both objects be the same?
  46. D

    Calc derivatives - Minimum total surface area in a box of V = 160 ft2

    Hi everyone! I'm new to online math forums. I wonder if anyone can give me a hand on this - it would be greatly appreciated. Thank you in advance! Dave Homework Statement If a box with a square base and an open top is to have a volume of 160 cubic feet, find the dimensions of the box having...
  47. S

    Polar Regions: Area, Arc Length, and Surface Area

    Homework Statement Consider the graph (see attachment) of r = 1 +2cos\Theta in polar coordinates. SET UP integrals to find 1. the area inside the large loop minus the area of the small loop. 2. the arc length of the small loop 3. the surface area of the surface formed by...
  48. C

    Center of Mass and Surface Area (multivariable)

    Homework Statement 1. Verify the given moments of inertia and find the center of mass. Assume each lamina has a density of p=1. The problem gives a circle with a radius a. 2. Find the area of the surface of the portion of the sphere x^2 + y^2 + z^2 = 25 inside the cylinder x^2 + y^2 = 9...
  49. T

    Surface Area Problem: Rotating y=x^4/16+1/2x^2 About Y-Axis

    Homework Statement Find the surface area obtained by rotating the curve y = \frac{x^4}{16} + \frac{1}{2x^2} 1 < x < 2 about the y-axis. The attempt at a solution http://img341.imageshack.us/img341/3245/mathy.jpg My final answer is \frac{41\pi}{10}. Is this correct?
  50. O

    Surface Area of a Solid of Revolution

    Homework Statement Homework Equations integral [a, b] 2*pi*x*sqrt(1 + (dy/dx)^2) The Attempt at a Solution Well I know how to do surface area questions... But that the @#$@ is with this random equation? How would I even start to evaluate it... Like honestly... I don't even understand the...
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