Area Definition and 1000 Threads

Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).
The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.
Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

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  1. J

    Green's theorem and area calculation

    In wiki there is the follows formula: https://en.wikipedia.org/wiki/Green%27s_theorem#Area_Calculation But, I don't understand why M = x and L = -y. I don't found this step in anywhere.
  2. D

    MHB Surface Area of ball floating in water

    Using calculus, how would I derive a formula for the exposed surface area of a ball floating in water? For such a formula to be a good candidate, it would have to consider oscillations of the water and placid water. The surface area of a sphere is \(A = 4\pi r^2\).
  3. B

    Vector Methods To Find An Area

    Hello, after searching around on the internet about this problem, it looks like it is Cramer's rule that I want to use, though it wasn't shown to us under that name. My textbook doesn't cover the material required for this problem, so i'd really like to run what I have done past you guys...
  4. Y

    MHB Area Under Curve: Finding the Missing Area

    Hello, I am looking for the area between \[f(x)=x\cdot ln^{2}(x)-x\] and the x-axis. I have a solution in hand, it suggests that the area is: \[\int_{\frac{1}{e}}^{e}(x-x\cdot ln^{2}(x))dx\approx 1.95\] I have a problem with this solution, I don't understand where the area between 0 and...
  5. M

    Izod Impact test Cross Sectional Area

    Homework Statement Ok, so in a lab I was preforming Izod impact tests on notched polymer specimens. To complete my calculations I need to determine the correct cross-sectional area to use (Which is baffling me as the simple things usually do) The sample was loaded with the notch facing...
  6. G

    Rate of change of area of circle in respect to radius

    What is the Rate of change of area of circle in respect to radius when radius is 3in I know that that dA/dr is equal to the circumference of the circle But where does that come from? Also the formula for the circumference of the circle is 2(pi)r But the answer is 6 (pi)in^2/in. I understand...
  7. T

    MHB Help me cut the area under a parabola in half

    The problem is: Consider the area under the curve f(x)=2x-x2 and above the x axis. Find the equation of the line through the origin that cuts this area into two equal parts.
  8. aboojoo

    Difficulty understanding formula and area under v-t graph

    1. The problem For sake of format I attached the a screenshot of the course material I'm having difficulty wrapping my walnut around. Which is how: Total Displacement = Area of Triangle + Area of Rectangle or Δvector d = Atriangle + ARectangle or Δvector d = 1/2 (V2-V1)Δt +V1*Δt...
  9. P

    Mean Aerodynamic Chord and Area

    If I have a problem in which the laminar/turbulent transition point is said to be 50% the mean aerodynamic chord, how can I find the area of the wing over which there is laminar flow? Is it simply half the wing area?
  10. anemone

    MHB Find the area of an equilateral triangle

    Show that the curve $x^3+3xy+y^3=1$ has only one set of three distinct points, $P$, $Q$, and $R$ which are the vertices of an equilateral triangle, and find its area.
  11. S

    MHB Understanding the Area of a Curve: A^4 Y^2 = X^4 (A^2 - X^2)

    I have to find the area of the loop of the curve a^4 y^2=x^4(a^2-x^2). I have confusion regarding the shape of the graph the limits of integration.
  12. A

    MHB Stuck trying to integrate the surface area of a curve

    Here's the problem I was given: Find the area of the surface generated by revolving the curve x=\frac{e^y + e^{-y} }{2} from 0 \leq y \leq ln(2) about the y-axis. I tried the normal route first... g(y) = x = \frac{1}{2} (e^y + e^{-y}) g'(y) = dx/dy = \frac{1}{2} (e^y - e^{-y}) S = \int...
  13. aleksbooker

    Can't integrate the surface area of revolving curve the normal way

    Homework Statement Find the area of the surface generated by revolving the curve x=\frac{e^y + e^{-y} }{2} from 0 \leq y \leq ln(2) about the y-axis. The Attempt at a Solution I tried the normal route first... g(y) = x = \frac{1}{2} (e^y + e^{-y}) g'(y) = dx/dy = \frac{1}{2}...
  14. N

    MHB Area under curves and Limit of a sequence,

    Hello, I am looking for an help about this, I have very short time to do many of them and those are an example, could someone show me one solution or explain me how to do it? Thank you if you can help me, I really appreciate. Francesco.
  15. Philosophaie

    Geodesic Equation & Orbital Surface Area Around the Sun

    The "s" in the geodesic equation refers to the "surface area" for that portion of the orbit around a star or black hole. For a small enough "delta t" the surface areas are the same. Around a small star the orbital surface area (without the other interfering gravitational sources) would...
  16. T

    Integral by interpreting it in terms of area

    Homework Statement ∫(a= -3 , b= 0) (1 + √9 - x^2) dx Homework Equations ∫(a,b) f(x) dx = lim as n → \infty \sum f(xi) delta x The Attempt at a Solution I tried plugging in my a and b value into the function just as I would with any other function to find the area and i get a number...
  17. A

    Find area under curve and energy of function (matlab)

    Homework Statement For the 1st one you wouldn't really need MATLAB I guess to find the area under the curve, it is 0 and so is its energy. For the 2nd one I got A=1.73 and so E=2.99.Homework Equations area under curve = evaluate integral from t=t1 to t=t2. in this case t=-2 to t=5 since they...
  18. S

    MHB Finding Area Bounded by x^2 & 2x - x^2

    Find the area bounded by the curves y = x^2 and y = 2x - x^2so x^2 = 2x - x^2 2x - x^2 - x^2 = 2x - 2x^2 So then would I factor out a 2 and get 2x(x - 1) x = 1 So the \int ^1_0 Right - left \, dx
  19. S

    How Does Human Luminosity Compare to Skin Surface Area?

    Homework Statement A typical adult burns about 2500 Calories in one day. 1)How much energy does the average human emit every second 2)What is another term to use for expressing a “Joule per second”? 3)From your answers to (1) and (2), you just determined the “luminosity” of the average human...
  20. B

    Finding the Area Bounded by Curves | Quick Solution

    Hello, quick question really. Homework Statement Find the area bound by the x axis, x = 1, x = 4 and y = 2/x Homework Equations The Attempt at a Solution Representing this graphically, the question is equivalent to performing the definite integral of y = 2/x from 1 to 4. Right? Which...
  21. C

    Minimizing area of a semicircle bounded by two lines

    Homework Statement The figure shows a semicircle with radius 1, horizontal diameter , and tangent lines at and . At what height above the diameter should the horizontal line be placed so as to minimize the shaded area http://imgur.com/grrCqWF Homework Equations The equation of a...
  22. I

    Deriving the surface area equation

    consider the following image (the red is the surface area element and the green is the differential element that I'm integrating over) when we derived this in class, we treated the area formed by vectors a and b, as the area of a parallelogram. the thing is, a and b should be at right...
  23. S

    MHB Finding the area bounded by the curves

    So it's been a while since I've done one of these problems. Need to make sure I am using the right procedures to solve it. Q)Find the area bounded by the curve $y = \frac{1}{2}x^2$ and $x^2 + y^2 = 8$ So first thing I did was plug in numbers to get the two graphs. It looks like they intersect...
  24. A

    How do I calculate the number of collisions per unit area in an ideal gas?

    I an ideal gas how do I calculate the number of collision per unit area? By collision I do not mean collision between the atoms but rather it is a problem where I know that a nucleation cluster of area A is in my gas, and I want to find the probability that it will get hit by an atom. I know the...
  25. MarkFL

    MHB Area of a Cardioid - Littlehime's Question @ Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can see my work.
  26. C

    What is the surface area when a curve is rotated about the x-axis?

    Homework Statement Obtain the surface area when the curve y=ex, 0≤x≤1, is rotated about the x-axis Homework Equations Surface Area = 2∏a∫b x√(1+(dy/dx)2)dx The Attempt at a Solution I started with the the equation, Surface Area = 2∏0∫1 x√(1+e2x)dx. However, whichever way I try to...
  27. C

    Surface area from bands: Calculus

    Homework Statement The shaded band shown here is cut from a sphere of radius Rby parallel planes hunits apart. Show that the surface area of the band is 2piRh. The image is on this site: http://imgur.com/TCx1weD http://imgur.com/TCx1weD The Attempt at a Solution How do I do...
  28. MarkFL

    MHB Minimizing Lateral Area of a Cone with Fixed Volume | Leprofece

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  29. I_am_learning

    Area of a general n-sided polygon

    Finding the area of an irregular polygon with n side is quite easy when we are given the length of all of the n sides and the length of (n-3) specific diagonals. This way, we get (n-2) triangles whose areas can be calculated using Heron's formula and then added up. But what if the length of...
  30. J

    Fanno flow constant area duct Re number

    I am trying to calculate the friction factor for a Fanno flow in a constant area duct. I know the friction factor is based on Reynolds number however does the Reynolds number not vary along the pipe due to a change in dynamic viscosity? (caused by a decrease in temperature along the pipe). And...
  31. D

    Area of a Parametrized Surface

    Here's my work: http://i.imgur.com/UMj72Ub.png I used the surface area differential for a parametrized surface to solve for the area of that paraboloid surface. My friend tried solving this by parametrizing with x and y instead of r and theta which gave him the same answer. I would greatly...
  32. IridescentRain

    Mathematica How do I calculate an area in Mathematica?

    Hi! Say I have a region described by any number of inequalities. This region is a surface in 3D space. How can I ask Mathematica to calculate the region's area? If it helps, my particular region is the intersection of a hollow sphere and a solid (i.e. filled-out) toroid-like surface. I'm...
  33. MarkFL

    MHB Kuba's question at Yahoo Answers regarding finding the surface area of a torus

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  34. L

    Area under v vs t sinusoidal trace

    if I have a sinusoidal trace on an oscilloscope (v vs t) and I wanted to find the area under the wave form squared graph I could integrate the sqaured waveform with respect to t. but since i don't have the integration facility... is it fair to say that the area under the graph is proprtional...
  35. ShayanJ

    Volume and Area in physical equations

    In some parts of the physics,sometimes it happens that the volume of a region of space or the area of a surface enters into a formula.In such situations,most of the time,the author argues that "although I have derived this formula for such a shape,it is independent of the shape of the...
  36. B

    Area element vector for parametric surface integrals

    When doing surface integrals of surfaces described parametrically, we use the area element dA = ndS = (rv x rw)dvdw Where dS is the surface area element and v and w are the parameters. I'm fine with the derivation of this (I think) but I don't understand why it's necessary to have n and dS...
  37. J

    Is computational physics a PhD area of study?

    If someone wanted to go into computational physics, would that be the PhD area of study, or is it just a branch of the better known areas, like hep, condensed matter, etc. Also is it considered theoretical or experimental? (I would assume experimental). Thanks!
  38. Albert1

    MHB Area of Circle O: Calculate Here

  39. M

    Finding cross-sectional area from terminal velocity.

    Homework Statement The aerodynamic drag on an object moving through air is proportional to Av^2, where A=cross-section, and v=velocity. The terminal velocity of a person without a parachute falling through the air is about 56m/s. Estimate the area of the cross-section of a person seen from the...
  40. MarkFL

    MHB Compute Area of 2 Simultaneous Inequalities: Don Leon at Yahoo Answers

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  41. E

    Focusing on a specific area of Mechanical Engineering Undergrad?

    I am majoring in Mechanical Engineering, and I found a research opportunity I really want (it focus on Fastening and Joining). I want to take classes relevant to this area, but my university offers different areas to specialize in as an Undergrad, so which one should I focus on in order to take...
  42. Saitama

    MHB Calculating the Area Between Two Curves: A Shorter Method?

    Problem: Calculate the area of region defined by the inequalities: $$-1<xy<1$$ $$-1<x^2-y^2<1$$ Attempt: Although I have solved the problem but I am not very satisfied with the method I used. The graph of region is symmetrical in all the four quadrants so I calculated the area in the first...
  43. ShayanJ

    Choosing the research area in physics

    I want to know how people usually choose their area of research?Is it just interest?What about choosing between areas which are all of your interest? I know,it may seem weird for an undergrad to ask that,but I think its the time for me to decide on it.Because I started studying physics when I...
  44. T

    Really Challenging Area Between Curves Problem

    Homework Statement Suppose C:y=f(x) with f a twice-differentiable function such that f''(x)> 0 for each x on the closed interval [0,a] where a is a positive constant. Suppose T is the tangent line to C at a point P= (r,f(r)) on C where r is in the open interval (0,a). Let A be the area of the...
  45. MarkFL

    MHB Bari's question at Yahoo Answers regarding minimizing the surface area of a silo

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  46. A

    Negative area between two curves

    I've been trying to figure out what a negative area means, but I can't. Homework Statement Calculate the area between f(x) = 3^{x} \, , \, g(x)=2x+1 The attempt to a solution The intersections are located in x=0 and x=1. So I do the integral from 0 to 1. \int_{0}^{1} (g(x)-f(x))dx =...
  47. MarkFL

    MHB Find Area of Lemniscate Bounded by Circle: r^2=6sin(2theta) & r=sqrt(3)

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  48. M

    Surface Area of Solid: f(x)=1/x, [1,∞) about x-axis

    Homework Statement f(x) = 1/x Interval [1, ∞) about the x-axis Set-up the integral for the surface area of the solid Then use the substitution u = x2 and integrate using the formula: ∫ sqrt(u2 + a2) / u2 du = ln(u + sqrt(u2 + a2) - sqrt(u2 + a2) / u + C a is a constant Homework...
  49. MarkFL

    MHB Kylie's question at Yahoo Answers regarding minimizing the area of a triangle

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  50. K

    Alternative way to calculate the area of a right angled triangle

    Hello all! New to the forums, and I have a question for you. In my classes, we have been dealing a lot with proofs lately, so when I was working on an assignment, I figured I would try and find my own proof for something, just for the hell of it. I decided to tacle the area of a right angled...
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