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. E

    Optimizing surface area of a silo

    A silo has cylindrical wall, a flat circular floor, and a hemispherical top. If the cost of construction per square foot is twice as great for the hemispherical top as for the walls and the floor, find the ratio of the total height to the diameter of the base that minimizes the total cost of...
  2. S

    Finding radius from volume that gives the minimum surface area for cylindrical cans

    i need MAJOR HELP! this is the problem: The can do tin can company minimizes costs by construckting cans from the least possible amount of material. this company suppplies many different sizes of cans to packing firms. a designer with the company needs to determine the radius that gives...
  3. I

    Surface area of functions without definite integrals

    Homework Statement The curve is rotated about the y axis, find the area of the resulting surface. y=(1/4)X2-.5ln|x| 1<_X<_2 Homework Equations S=2(pi)(f(x))\sqrt{}1+f'(x)^2 The Attempt at a Solution Alright I'm not entirely sure where to even begin. Since I'm rotating about the Y-axis I know...
  4. R

    Surface area vs rate of evaporation

    Two vessels of the same volume, in the same location and open to the atmosphere. One is tall and of a small diameter, the other is short and is of a larger diameter. Each vessel supplied with an identical heat source. Each vessel filled with an equal volume of an identical liquid. The goal...
  5. T

    Electric field due to a line charge, surface area

    Homework Statement A charge distributed uniformly over 1 face of the circular disk, find the surface charge density ? well i know density for surface charge is S= charge/area, the area for face of 1 circular disk is what ? 2pir ?
  6. A

    Surface area of intersecting cylinder and sphere

    I have a problem that I've been stuck on for a while as follows, Find the surface area of the part of the cylinder x^{2}+y^{2}=2ay in the first octant that lies inside the sphere x^{2}+y^{2}+z^{2}=4a^{2}. Express your answer in terms of a single integral in \phi, you do not need to evaluate...
  7. A

    Finding D: Solving for Surface Area of Function

    Homework Statement I have this function, and I need to find the Surface area of the function that is confined between the plains x=0, y=0 and z=0. My question is, what's D? Homework Equations The Attempt at a Solution
  8. F

    What is the Surface Area of a Parametric Surface in the UV-Plane?

    Homework Statement Find the area of the surface with parametric equations x=uv, y=u+v, z=u-v. u^2 + v^2 <= 1Homework Equations A(S)= double integral over the domain D of the norm of the cross product (r_u X r_v) DAThe Attempt at a Solution I started off with finding the norm of the cross...
  9. J

    Optimization: Minimum Surface Area

    Homework Statement I need help on an optimization problem involving a hexagonal prism with no bottom or top, but the top is covered by a trihedral pyramid which has a displacement, x, such that the surface area of the object is at a minimum for a given volume. The assigned variables include...
  10. J

    Surface Area of 3d graph

    Homework Statement Find the area of the sphere x^2 + y^2 + (z-2)^2 = 4 that lies inside paraboloid z = x^2 + y^2Homework Equations The Attempt at a Solution when i take the equation of the spere and replace x^2 + y^2 with z i get: z(z-3) = 0 so they intersect at the plane z = 3. were supposed...
  11. L

    Arc length/ surface area with integrals

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  12. rohanprabhu

    What is the true surface area of the Earth according to General Relativity?

    I have a very faint idea of General Relativity.. hence this question. I think that according to General Relativity, the shortest distance between two points on this Earth is along a curved path, which is the curvature of the Earth [or sorta.. parallel to it]. Hence, i assumed that when on earth...
  13. C

    Surface Area Equality: The Simplest Explanation

    ok why does surface area = ( y) ( 1+ f ' (x)^(2) )^(1/2) equal surface area = (x ) (1+ f ' (x)^(2) )^(1/2) the simplest explanation please.
  14. E

    Arclength, Surface Area and Volume

    I was just thinking: If \iint dS is the surface area of a level surface, S, and \iiint dV is the volume of an enclosed solid, V, shouldn't \int df be the arclength of a function f(x)? Lets say that our surface is given implicitly by \Phi For the surface area we get: \iint dS =...
  15. I

    Surface area of a cone problem

    Homework Statement The question is to derive the surface area of a cone. Homework Equations slant= square root ( r^2 + h^2) surface area= int int [square root(fx^2 + fy^2 +1) da] surface area of cone side= pi *r(r^2+h^2) 3d cone formula: z= h/r(squareroot x^2+y^2) The Attempt at...
  16. C

    Express the surface area of a cube

    Homework Statement Express the surface area of a cube as a function of its volume. Homework Equations Cubic Volume=Length x Width x Height (V=Length of side^3) Cubic Surface Area= (Length of side^2)x6 The Attempt at a Solution f(V)=(X^3/X) x 6...sorry, I don't know if I'm on the...
  17. I

    Surface area of a polar equation

    Hello, the problem I'm working on is to find and set up the integral whose value is the area of the surface obtained by rotating the curve about the x-axis, then another integral to find the surface area by rotating about the y-axis. I do not need to evaluate these integrals, just set them up...
  18. T

    Surface area and parametric equations

    I just have a question, when I am rotating something let's say around y=2 and the two equations are x=t^3 + 1 and y = 4t+1 how would i set it up?
  19. S

    Find Surface Area obtained by rotating a curve?

    Find the area of the surface obtained by rotating the curve y=2e^(2y) from y=0 to y=4 about the y-axis. Any help on this would be greatly appreciated. This has my whole hall stumped. We know that you have to use the equation 2pi*int(g(y)sqrt(1+(derivative of function)^2), but cannot figure...
  20. K

    What is the surface area of a Mobius strip made from a strip of paper?

    [SOLVED] Mobius Strip we have a normal strip of paper with surface area=A. if we make a mobius strip with it what will be the area of the mobius strip? is it A or 2A?
  21. D

    Surface area problem in 3-d calculus

    Homework Statement Find the area of the surface. The surface z = (2/3)(x^(3/2) + y^(3/2)), 0 </= x </= 1, 0 </= y </= 1 Homework Equations Double integral over S of the magnitude of dr/du cross dr/dv dS, which equals the double integral over D of the magnitude of dr/du cross dr/dv...
  22. S

    Optimizing Can Dimensions for Minimizing Material Usage

    [SOLVED] Minimizing Surface Area Homework Statement A can is to be manufactured in the shape of a circular cylinder with volume = 50. Find the dimensions of a can that would minimize the amount of material needed to make the can. Homework Equations V = \pi r^2 h SA = 2 \pi r^2 + 2...
  23. G

    What is the Relationship Between Volume and Surface Area of Revolution?

    The Volume of a revolved function can be given by the integral of pi*f(x)^2*dx. For the arc length of a graph, a different integral is available. I understood the proof of these two and their integration is understandable. From such, I was actually expecting that the surface area of...
  24. U

    What dimensions minimize the surface area of a box with a fixed volume?

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  25. R

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  26. E

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  27. R

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    Dear all, I was reading through my notes and I was kinda like stumbled in the way the minimum surface area of a cylinder has been derived. First, A= 2*PI*r^2 + 2*PI*r*h and given the condition that the volume has been fixed, the resulting area equation becomes A= 2*PI*r^2 + 2V/r...
  28. P

    Why a magnetic flux in closed surface area is always 0?

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  29. G

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    The given surface area of a sphere is 4*(pi)*r^2. There are several proofs to this, but I'm just looking for the error in this one: For the arc length, s = rx, x being the angle in radians. Therefore, s = 2(pi)r = C, C being circumference of circle. The surface are of a sphere is the sum...
  30. A

    Surface Area of Plane Inside Cylinder: Solved

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  31. H

    Solvent accesable surface area

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  32. M

    Surface area of a sphere-derivation

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  33. C

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    Am I correct in saying the surface parameterized by r = (sin v, u, cos v), v = [-pi/2, pi/2], u = [-1, 1] has an area of 2pi ? I get something different by computing the arc length of the parabola within the bounds and multiplying by 2. Which method is wrong?
  34. N

    Using integrals to get volume, center of mass, and surface area

    Homework Statement For the homogeneous ice-cream cone that is given in spherical coordinates by rho= pi/4 (the bottom part) and phi=cos(rho) (the top part), find the volume, the center of mass, and the surface area. ((You have to do this problem using integrals, known formulas from...
  35. S

    Finding Dimensions of Cone with Surface Area 1 and Max Volume

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  36. M

    Surface Area and Surface Integrals

    Homework Statement (Q) Find the area of the surface cut from the paraboloid x^2+y+z^2 = 2 by the plane y=0. Homework Equations The Attempt at a Solution The unit normal vector in this case will be j. Moreover, the gradient vector will be sqrt(4x^2+4z^2+1). And the denominator...
  37. M

    Surface Area and surface Integrals

    Homework Statement (Q) Find the area of the portion of the surface x^2 - 2z = 0 that lies above the triangle bounded by the lines x = sqrt(3), y = 0, and y = x in the xy-plane. Homework Equations The Attempt at a Solution The know how to find the gradient vector. The part...
  38. P

    How to find the surface area of a spherical triangle?

    Hello I have a spherical triangle with the radius 1, and I have tried so hard to find the surface area. I know that A=120°, b=90° and c=60°. I could calculate that B=73.89° and C=56.31° and a=115.66°. I think I should use the formula (ABC) = (A + B + C - pi) r2 I always get the...
  39. 5

    Useable surface area for a throttle body (valve), is this correct?

    Here is some information I have for throttle bodys: 60 millimeters is equal to 2.362205 inches = 17.52 square inches 65 millimeters is equal to 2.559055 inches = 20.56 square inches - (17%) 70 millimeters is equal to 2.755906 inches = 23.85 square inches - (36%) 75 millimeters is equal to...
  40. A

    What is the Formula for Calculating Surface Area of z=x^{2}+2y in a Given Range?

    I need to find the surface area of z=x^{2}+2y where 0\leqx\leq1 and 0\leqy\leq1. I figured it's like trying any other surface area problem, but I think I'm misunderstanding how to set up this problem. Here is what I tried: \int^{1}_{0}\int^{1}_{0}\sqrt{2x+3}dydx = \frac{5\sqrt{5}}{3}-\sqrt{3}...
  41. S

    Surface area of N spherical droplets?

    Homework Statement I have the following problem Assume that 30.0 cm^3 of gasoline is atomized into N spherical droplets, each with a radius of 2.00 x 10^-3 m. What is the total surface area of these N spherical droplets? Homework Equations SA = 4 * pi *r^2 V = 4/3 * pi * r^3The Attempt at a...
  42. S

    Surface area of N spherical droplets?

    I have the following problem Assume that 30.0 cm^3 of gasoline is atomized into N spherical droplets, each with a radius of 2.00 x 10^-3 m. What is the total surface area of these N spherical droplets? I calculated the surface area of each atom to be 5x10^-9 m^2. I also calculated the volume...
  43. N

    Lightyears and the surface area of a planet.

    A light year is the unit of distance that light, with a speed of 2.99792e8 m/s, travels in one year. What is the surface area of a planet whose radius is 1300 km? Answer in units of lightyears^2. I am wondering if my method is correct. SA of a sphere is 4[pi]R^2. With R = 1300, the...
  44. R

    Deriving the Surface Area of Sphere

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  45. R

    Surface Area of Satellite Signal transmission on Earth

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  46. J

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    This is what i have so far SA = 2(pi)r^2 + 2(pi)r(245.45 / (pi)r^2) Derivative of SA = - 490.9r^2 + 4(pi)r 0 = -490.9r^2 + 4(pi)r and I'm stuck at this step.. I've tried a bunch of things to try and solve for r but i can't seam to get a logical answer for the radius of the minimized...
  47. K

    Calculate Blue Surface Area of Circles and Rosette

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  48. A

    Iterated Integral Surface Area Problem (with Polor Coordinates)

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  49. Z

    Surface Area Multiple Integrals problem

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  50. W

    Maximum eror in calculated surface area

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