Areas Definition and 246 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. A

    Good areas in math where I can make some contributions?

    I know this sounds like a weird question, but I'm interested. So I've always loved math, especially pure mathematics. I spend a lot of time reading about theorems, mathematical proofs, and I try to come up with my own proofs. Recently I had the idea that maybe I can spend my time reading some...
  2. Andreol263

    What math areas I need to see for graduate-level physics?

    I'm have already learn some of Differential,Integral,Multivariable Calculus(with Vector Calculus to), Ordinary Differential Equations, and something in PDEs(i know solve some by the Method of Separation of Variables:woot:), so what should i need to learn in math for areas like Eletromagnetism...
  3. C

    Exploring Unrelated Areas of Study: Why and How?

    Hello all, I was hoping some of you who have taken or are currently taking courses in an area unrelated to your major would share something about the experience: What are the two subjects, and why did you choose to pursue formal education in the second area of study? Did you take only a few...
  4. B

    Volume of a tetraedron in function of the areas

    Given any tetraedron, I want to calculate the volume V in function of the areas of the surfaces of the solid. I found this pdf that explain this: http://daylateanddollarshort.com/mathdocs/Heron-like-Results-for-Tetrahedral-Volume.pdf But, o pdf says that beyond of the 4 faces (X, Y, Z, W) is...
  5. P

    Bacteria only found in dry areas as cysts?

    Hi! I'm doing a worksheet on Body Size, Heat Loss and SA/Vol Ratio. The question I'm confused by is: "The smallest organisms such as bacteria, protozoa and unicellular algae live in water or a saturated atmosphere. They are only found in dry situations as cysts or spores. Why?" The wording...
  6. AdrianHannon

    Engineering Info on weapons engineering and related areas

    Hi, I'm in first year of my physics undergrad degree in NUIGalway and currently (could change as the four years pass) have an interest in doing a masters in Weapons Engineering after the degree is completed. In my course we get the choice of specializing in one of four areas: theoretical...
  7. F

    Find Area Under x=2Sin^2(y) & y=x^2 Graphs

    Find the area of the regions shown in the figures. These are the graphs used : y = x^2 x = 2 Sin ^2 (y) I know that I need to set the two equations equal to each other in order to find the points of intersection, but I run into some trouble when trying to simplify it for y. This is what...
  8. Bassirou

    Research Areas for PhD in Physics & Industry R&D

    Which research areas for a PhD in Physics may lead to good positions in the Industry's R&D? Please, excuse my English and thank you for any suggestion.
  9. A

    Engineering Areas of electrical engineering with the most potential

    I'm currently a second-year electrical engineering major (in the US). Next semester I'll begin to take almost exclusively classes involving different branches of electrical engineering. I've taken two digital design courses, a course on microprocessors, and I'm currently on my second "circuits...
  10. T. Wentling

    What background fits promising areas of mathematical physics

    I'm a graduate mathematics student and I did my undergrad in applied math. I also took the normal 10 hrs of physics foundations and then a semester of modern physics (basic quantum intro, special relativity, orbit states etc.). I was thinking about pursuing study in areas that would be...
  11. N

    Current Areas of research in Optics?

    Hello, I'm not sure whether this thread should be in a guidance section or here, but I think its more closely related to this area. As the title of the thread says, what are some current areas of research in Optics, if any? Thanks for any and all replies
  12. samgrace

    Which areas of maths are from which fundamental areas

    Hello, I am a physics student and have catagorised most of physics, e.g classical mechanics, relativistic mechanics, quantum mechanics and quantum field theory, and have also identified all the mathematics involved in each of these catagories. For example classical mechanics involves Calculus...
  13. 1

    Research areas in condensed matter physics

    What are some exciting research areas in condensed matter physics at the moment?
  14. PhysicsKid0123

    Surface integrals/Surface areas of arbitrary domain regions

    I'm having trouble evaluating this surface integral. This would be very simple to solve if the parameter domain of the variables u and u was a square region. However, that isn't the case here. I've tried using a change of variables and saying that u = r cos x, and v = r sin x. Where 0 < x < 2pi...
  15. W

    What are the main areas of physics and mathematics for aerospace engineer?

    Hello. I'm not a mechanical or aerospace ingineer, but it's interesting for me. What are the main areas of physics and mathematics for aerospace engineer? It seems to me, that main areas of maths are calculus and differential equations, maybe some linear algebra. And I think, that main areas of...
  16. R

    Areas Of Reasearch in Fluid Mechanics & Mechanical Design

    Hello! My friend and i are interested in doing a project.My area of interest in Fluid mechanics and my friend's is mechanical design can you guys help us out by providing some topics of research which encircles both these areas.
  17. P

    So do you think using modern physics creates new areas in mathematics?

    Hello people, I just joined the physics forum and it's great to see that they're other physics enthusiasts on the internet! The reason why modern physics is appreciable for me is that it creates new areas in mathematics like this for example brought up from Michio Kaku... "String theory...
  18. W

    Models for Determining Volumes and Surface Areas

    When we generate solid by rotating a curve around an axis, we use "slabs" of cylinders to approximate the volume of this solid of revolution. When we want the find the surface area, we instead use "slabs" of conical frustums (ie. the slope of the differential length of curve is taken into...
  19. J

    Areas of a series of annular sectors

    Hi all Hoping someone can figure out a problem. In the attached figure, A is the area of openings in a disc. Assume the segments 82, 84, 86, 88 are along a radius and are equal. Is it possible to prove that A4/A3 > A3/A2 > A2/A1? In thanks I'll share this, hope it's not old news. Some...
  20. E

    Areas of EE that use computations and modeling?

    I'm currently in the middle of a Physics PhD program but I now want to drop out with a Masters. I'm considering a career as an materials engineer or EE, but I definitely want to focus more on using software than 'hands-on' work. I heard that power systems, signal processing, communications...
  21. A

    Is the efficiency of heat engine more in hilly areas than in plains?

    In hilly areas temp is low than plain areas.So the temperature of source as well as sink must be low.But ratio remains same.So efficiency is same as in plain areas? Is my logic correct? Or anything else? Can it be more?
  22. R

    How Do You Solve the Areas Between Curves Problem Involving Unknown Functions?

    Homework Statement Let a > 0 be a fixed real number. Define A to be the area bounded between y=x2,y=2x2, and y=a2. Define B to be the area between y = x2, y = f(x), and x = a where f(x) is an unknown function. a) Show that if f(0) = 0, f(x) ≤ x2, and A = B then int 0-->a2 [y1/2-(y/2)1/2] dy =...
  23. srfriggen

    Recreational problem involving circles, cords, and areas

    Hello all, I've been trying to work out this problem I came across yesterday when a professor mentioned it in a math education course. It states, "Take a circle and put two dots on the circle then connect them with a cord. How many sections of area does the cord split the circle into?" Of...
  24. B

    Integral Question: Areas between curves?

    Homework Statement Sketch the region in the first quadrant enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region. y=4cosx, y=6sin2x, x=0. Homework Equations The Attempt at a Solution I got the intersections, which...
  25. N

    Exciting new areas of research?

    Hey! I am going to choose my specialization soon in my engineering physics degree, but I am very uncertain on what I want to do. Only thing I know is that I want to work on something cutting edge. So what areas of science and tech do you find exciting, and why? Do you know anything about...
  26. J

    Polygons: Areas Homework Help & Hints

    Homework Statement (I have attached the problem to this post as a file) Homework Equations In class we learned other fomulas for the area of a triangle using the SAS case, ASA and SSS (Heron's). The Attempt at a Solution I am honestly so confused with this one. I have a bunch of random...
  27. Hunus

    Euclid's Elements - The Application of Areas

    In Heath's commentary on Euclid's Elements he stresses the importance of the application of areas (Book I Proposition 44) with, "The marvellous ingenuity of the solution is indeed worth of the 'godlike men of old'...". The proposition, "To a given straight line to apply, in a given...
  28. R

    Application-oriented areas in Optics & Photonics

    Hello all, I am considering applying for PhD programs in Physics/EE (specifically - Optics/Photonics). Considering pervasive posts about the dismal job market, I understand Optics/Photonics is more applied, and hence more sought-after in industry. However, what areas within Optics/Photonics...
  29. U

    RNA destroyed, and areas of no color showed

    1. The problem in attachments Homework Equations 3. RNA destroyed means no proteins (enzymes) formed, however if some areas showed no color, does that mean other areas did? I'm pretty sure the answer is D but making sure
  30. MarkFL

    MHB Cfm4life's questions at Yahoo Answers regarding surface areas of solids

    Here are the questions: I have posted a link there to this topic so the OP can see my work.
  31. C

    Physics for Chemists: Which areas are of help to chemists

    Hi, I just finished my bachelor's degree in chemistry, in which I have made sure to get some extra mathematics and statistics (introduction to calculus, linear algebra, and linear differential equations; experimental design, ANOVA, advanced regression). I feel pretty covered here and ready for...
  32. K

    Areas Bounded by Trigonometric Functions.

    I will do my best to describe the problem I am working on. The problem is not from a textbook or anything but something I am working on independently to strengthen my first year calculus knowledge. What I did is I took sin(x) and -sin(x) and graphed them together. Sin(x) and -sin(x)...
  33. karush

    MHB Triangle and circle length and areas

    the following diagram shows a circle with center O and a radius 4cm The points A, B, and C Lie on the circle. The point D is outside the circle, on (OC) Angle ADC=0.3 radians and angle AOC=0.8 radians (a) find AD I used law of sines \frac{4}{\sin{0.3}}=\frac{x}{\sin{0.8}} x \approx...
  34. I

    Area Under Graphs: Finding Work Done

    Okay, so I've got a question on graphs and areas under curves. For example, a force-distance graph, with force on the y-axis and distance on the x-axis, where you find work done from that graph. I understand that a constant force will produce a horizontal line on the graph, like so: Force...
  35. J

    Statics, centroids of lines, areas and volumes

    Hi to everybody. I´m reading a book about statics and I cannot understand this chapter. I have been calculating moments of forces in hundreds of problems, when I found a force acting on a body I needed to fix a coordinate system, then calculating the moment arms of that force around a point...
  36. N

    Symmetric Converging-Diverging Nozzle Areas?

    Hey guys I was wondering if a symmetric C-D nozzle has the same area at the inlet and exit, can the mach numbers at both points be the same? This situation would involve a normal shock which would be near the exit as well. I would assume the Mach number at the throat would be 1. With that...
  37. MartinJH

    Exploring Opportunities in Computer Science: What Lies Ahead?

    I have used the search to find topics on this but to no avail. If there are topics asking this question then please advise me. I am due to start a degree in computer science in September, here in the UK. It is a life long passion and a degree course I could have never of dreamed of studying...
  38. T

    What areas of math do you need to know in order to understand calculus

    What areas of math do you need to know in order to understand calculus? Let's assume all I have is up to a 6th grade education. Can you tell me what and how I might learn these things.
  39. N

    Improper integrals and canceling of areas

    When evaluating an improper integral and i get infinity minus infitiy when taking the limit. in what case do the areas cancel?
  40. R

    Van de graaf options to increase reliability in humid areas.

    Hi, I am trying to design an electrostatic generator that works well in high humidity. At the moment I'm looking at a van de graaf generator and covering the metal sphere with an insulator and only leaving a little bit exposed. Is this safe? And will it increase the maximum charge? My...
  41. N

    Best mini/micro Hydro power for poor areas

    Hey guys, I am wondering what is the best type of mini hydro power plant to produce between 1 and 5Kw of energy. From what I understand most micro generator needs high head, and they take advantage of the relief (since power=head*flow*g) I am looking at an area where the head is very...
  42. D

    How to Evaluate Integrals Using Areas: A Scientific Approach

    The Question Let f(x) = |x|. use areas to evaluate ∫(-1,x)f(t)dt for all x. use this to show that d/dx∫(0,x)f(t)dt = f(x) not sure hot to evaluate the integral using area when i don't know what f(t) is...
  43. S

    Water Flowing through U-tubes (Velocity Given Cross-Sectional Areas)

    Homework Statement http://www.aapt.org/physicsteam/2010/upload/2010_FmaSolutions.pdf See question 23 A' = 1/2A Homework Equations Av = A'v' The Attempt at a Solution I thought this was just a simple Av = A'v' problem which would lead to v' = 2v. But there is apparently more to it, as the...
  44. C

    Same Areas of Trapezoid within Triangle, why?

    Good day, while reading up on an elementary math study book, i have encountered that a proof is build upon the following (see attachment for the figure). Are the Areas CDE and BED really the same? I tried to calculate this from abstraction, not sure where I could have made a mistake.. g1 is the...
  45. H

    What areas of brain research should I get involved in?

    I'm doing double honours in Neuroscience, likely biochemistry as well. I have to find a thesis adviser for next year...I know everyone says "follow your heart" or something to that effect, but that's not what I'm looking for, not to impugn anyone will kind intent. Here's my interested in Med...
  46. H

    Evaluate the integral by interpreting it in terms of areas.

    Homework Statement from [-2,2] rad(4-x^2) Homework Equations The Attempt at a Solution I know its a circle and i get the equation to be y^2+x^2=4 and I believe it has to be divided into a circle and rectangle so the area of the rectangle i got to be 2 the circle i got to be...
  47. alane1994

    MHB Approximating Areas under Curves

    I am learning this right now, and I am having troubles with something. For regular partition, the formula in my textbook is this. x_k=a+k\Delta x, \text for k=0,1,2,...,n. My question is this, how does one find "k"? It is very important clearly!;)
  48. elementHTTP

    Protecting electronic circuits in hazardous areas ?

    I am interested in how to protect electronics circuit in hazard areas Are there special ways of soldering /circuit arranging for highly vibrational places ? moist/water protection ? Impact ? Hi temperature ? Special box materials for examples above? etc... tnx ;D
  49. RabbitWho

    Medical Brain terminology question - association areas?

    Müller's law seems to differ from the modern statement of the law in one key way. Müller attributed the quality of an experience to some specific quality of the energy in the nerves. For example, the visual experience from light shining into the eye, or from a poke in the eye, arises from some...
  50. J

    CFM calculation with a known fan and different inlet and outlet areas

    Hello, Here are the known's of a chassis: Inlet open area = 17 inches^2 Outlet open area = 30 inches^2 Internal Fans = 270 cfm (3 fans @ 90 cfm each) What is the CFM through the system and what's the equations used to figure this out? I assume that if we had a very small inlet or...
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