Can be shape of the Universe be defined using the Calculus "Washer Method" formula?
955258
You can use either method.
285273
The Universe IS Infinite !!! \n\nThe shape of the universe is determined by a struggle between the momentum of expansion and the pull of gravity. \nThe rate of expansion is expressed by the Hubble Constant, Ho, while the strength of gravity depends on the density and pressure of the matter in the universe. If the pressure of the matter is low, as is the case with most forms of matter we know of, then the fate of the universe is governed by the density. If the density of the universe is less than the "critical density" which is proportional to the square of the Hubble constant, then the universe will expand forever. If the density of the universe is greater than the "critical density", then gravity will eventually win and the universe will collapse back on itself, the so called "Big Crunch". However, the results of the WMAP mission and observations of distant supernova have suggested that the expansion of the universe is actually accelerating which implies the existence of a form of matter with a strong negative pressure, such as the cosmological constant. This strange form of matter is also sometimes referred to as the "dark energy". If dark energy in fact plays a significant role in the evolution of the universe, then in all likelihood the universe will continue to expand forever.  \n\n\nThe density of the universe also determines its geometry. If the density of the universe exceeds the critical density, then the geometry of space is closed and positively curved like the surface of a sphere. This implies that initially parallel photon paths converge slowly, eventually cross, and return back to their starting point (if the universe lasts long enough). If the density of the universe is less than the critical density, then the geometry of space is open, negatively curved like the surface of a saddle. If the density of the universe exactly equals the critical density, then the geometry of the universe is flat like a sheet of paper. Thus, there is a direct link between the geometry of the universe and its fate.
834202
Absolutely.\n\nThe terms "point," "plane," "straight line," and so forth are arbitrary terms used in mathematics. \nIn reality, the exact position of a point is completely dependent on the observer and his reference. A "starting point" will never match the "ending point" for all references after a certain amount of time has passed.\nRemember, there is no reference that is perfectly at rest relative to all other references - yet each will legitimately consider its own reference as stationary.
208438
There is a delightful article by the late Dr. John Kenny on that, back a few years ago, looking at the formulae for surface area and volumes of spheres; I'll see if I can find it for you--it's in my files SOMEWHERE--and give you the reference, and the formulae he generated. It turns out that they're not precisely the same as what you've got--but not terribly far from it.\n\nAs I said, I'll see if I can find it, and I'll get you the reference and the formulae.\n\nAbout 9pm CST...\n\nOK, here is the formulae, done the best I can on this thing...\n\n\nFor a sphere of dimension 2*k, radius R\n\nVolume: (pi^k)*(R^(2*k))/k!\n\nFor a sphere of dimension 2*k + 1 and radius R\n\nVolume:\n(2^(2*k+1))*k!*(pi^k)* (R^(2*k+1))/(2*k+1)!\n\n\nTo get the surface areas for each of these, all you do is differentiate with respect to R...\n\nAs I said, it was a little more complex than you thought--the surface areas come out a little bit differently for odd and even numbered dimensions.  I do not recall why, frankly.
347039
F = G*m1*m2/d^2\ninverse square law
459851
„input harvmac „overfullrule=0pt {„it Exam} (1) Prove via path \n„def„R{{„bf R}} „def„S{{„bf S}} (a) Work in four-dimensional (flat) Euclidean space $„R^4$ and show that conformal invariance implies...\nhttp://www.math.ias.edu/QFT/fall/exam1.tex \n\n \nhttp://www.math.ias.edu/QFT/spring/stringPS8.tex\n...def„det{{„rm det}} „def„Det{{„rm Det}} „def„tr ... B$ of the Wess-Zumino-Witten model, satisfy the equations for conformal invariance and for\nhttp://www.math.ias.edu/QFT/spring/stringPS8.tex \n\n \nComments: Latex(2e), 34 pages, 4 figures (eps) On some UNIX machines\nSome private shortcuts: „def„be{„begin{equation}} „def„ee{„end{equation}} „def„bea{„begin{eqnarray}} „def„eea{„end{eqnarray}} „def...\nhttp://www.ma.utexas.edu/mp_arc/e/96-436.latex.mpmix \n\n \n=4 19| >0 63| >1 65| >2 434| >4 525| 0 1773| 1 630| 1/2 2530| 10 \n\nhttp://www-library.desy.de/lists/spihep.k \n\n \nStardrive.org - - Article/Content Template\nStardrive.org is dedicated to fostering increased public awareness and education in the theoretical and technological aspects of interstellar travel.\nhttp://www.stardrive.org/Sarmail1-3-01.shtml \n\n \n„documentstyle[12pt]{article} %„renewcommand{„baselinestretch}{2.0}\n...oddsidemargin 0 mm „def„ee{„end{equation}} „def ... correlation functions have a power law behaviour, indicates that conformal invariance may be\nhttp://theory.ipm.ac.ir/papers/src/ipm63.tex \n\n \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%...\n...def „pa {„partial} „def „ti {„widetilde} „def „se {{„prime„prime}} „def ... „subsection{Pre-big bang inflation and conformal frames}\nhttp://www.ba.infn.it/~gasperin/tex/ba437.tex \n\n \nhttp://www.ba.infn.it/~gasperin/tex/cer96267.tex\n...def „pa {„partial} „def „ra {„rightarrow} „def ... discussed in „cite{6a}, to the conditions of $„sg$-model conformal invariance represented by\nhttp://www.ba.infn.it/~gasperin/tex/cer96267.tex \n\n \n%%%%%%%%% LATEX FILE FOR FORMATTING GUIDELINES \n...def„xp{X^{+}} „def„xm{X^{-}} „def„Yp{Y^+} „def„Ym ... must be obeyed (in a conformal gauge this is the same as maintaining conformal\nhttp://theory.tifr.res.in/group/colloq/das.tex \n\n \n„documentstyle[twoside,psfig]{article}\n...def„ds{„displaystyle} „parindent=0pt „def ... out that this model is advantageous for the study of the conformal invariance conjecture for\nhttp://www.math.utah.edu/~ejpecp/EcpVol1/paper8.tex
965826
L = 8pi G rho/c^4 is the definition of the cosmo constant L (1/m^2); where c = light speed and rho is the energy density of the known universe in a vacuum.\n\nF = GmM/r^2 is the equation for the attraction force F between two masses m and M, r distance apart.\n\nL was created by Einstein to make his equations come out right.  By that, he meant the universe should be spherical (not flat or saddle shaped).  His original results showed a flat universe, and he just "knew" that was wrong; so on came L to fix the shape problem.\n\nF was the result of experiments by Newton.  Through observation, he found that the attractive force between two bodies was proportional to the inverse square of the distance between them and to the product of the magnitudes of the two masses.  That is, F varies as (~) (mV/r^2).  \n\nWhenever there is a proportionality, as in F ~ mM/r^2, there has to be a constant of proportionality.  In this case Newton called it G; so that F ~ mM/r^2 becomes F = GmM/r^2 when the constant is inserted.\n\nThe G in both equations is the same constant.  But Sir Newton came upon it long before Einstein found it could be used in his equations and the calculation of L.  So there is no way that Newton (1642-1727) could know about Einstein's (1879-1955) use of it much later.
1033128
Rewrite the equation as y=9/x. If we take one point on that curve and rotate it around the axis, the circle generated has a circumference of 18π/x and an area of  81π/x² (2πr and πr², respectively). Each of these circles is as an infinitesimally thin slice of the entire solid of revolution, and so we can find the surface area and volume of it by integrating the circumference and area of these circles, respectively. Thus the surface area of the entire curve is (n→∞)lim (1, n)∫18π/x dx (that should be read as "the limit as in approaches infinity of the definite integral from 1 to n of 18 pi over x with respect to x"). The antiderivative of 18π/x is 18π ln (x) and thus the by the fundamental theorem of calculus, (n→∞)lim (1, n)∫18π/x dx= (n→∞)lim (18π ln (n) - 18π ln (1)). Since the limit of ln n as n→∞ is ∞, the surface area is infinite.\n\nRepeating for the volume: the volume is (n→∞)lim (1, n)∫81π/x² dx . Our antiderivative is then -81π/x, and so the final volume is (n→∞)lim -81π/n - (-81π/1). Since the limit of 1/n as n→∞ is zero, the first term in this expression is zero, thus the final answer is - (-81π/1) or simply 81π.
102944
I think these answers are skating around what you really want. First, I don't think you mean stretch, so that it fits any shape.\n\nYou mean to 'unfold' the sphere after cutting it. Your mental imagry is a good way to see how the area of a sphere is calculated, but we wouldn't do it in a physical form since the sphere would not flatten nicely. So those who argue that it can't be done are missing the real advantage of what this question gets at. At least in my mind.\n\n\nAfter much messing around with this, the shape, after all the unfolding, is similar to two parabolas joined together.\n\nThe problem is that because the parabola like shapes don't quite make a parabola, the formulas don't quite work out. The area of a sphere is 4*pi*r^2.\n\nIf we consider them parabolas each has area (2*(pi*r)^2)/3 making the whole shape area (4*(pi*r)^2)/3. Of course this doesn't quite equal 4*pi*r^2.  Pi is roughly 3.14 and the denominator is 3. If they were to cancel in the equation, it would be correct.\n\nBecause the shapes are close, but not actual parabolas, finding the area will require calculus in order to establish that it equals 4*pi*r^2.\n\nIronically, you could use the ancient method of exhaustion to arrive at the area. I think Archimedes did this, but the method was developed by a man with your name :).\n\nHey, being a math major is useful after all.
657544
Isn't the universe infinite? How can it have a definite radius or circumference if it's infinite?
747259
To find out whether there is an end or an edge to the universe we first need to know what is the shape of the universe.\n\nMost astronomers would like to know the shape of the universe too! There are three general possibilities. First, like your balloon, the universe might have what we call positive curvature, like a sphere. In this case, which we call a "closed" universe, the universe would be finite in size but without a boundary, just like the balloon. In a closed universe, you could, in principle, fly a spaceship far enough in one direction and get back to where you started from. Closed universes are also closed in time: they eventually stop expanding, then contract in a "Big Crunch." All the geometry that is true on a sphere is also true in a closed unvierse: parallel lines eventually converge (e.g. longitude lines are parallel at the equator, but converge at the poles), large triangles have more than 180 degrees, etc.\n\nThe second possibility is that the universe is flat. This kind of universe you can imagine by cutting out a piece of your baloon material and stretching it with your hands. The surface of the material is flat, not curved, but you can expand and contract it by tugging on either end. Flat universes are infinite in spatial extent, and have no boundaries. Parallel lines are always parallel and triangles always have 180 degrees. Flat universes expand forever, but the expansion rate approaches zero.\n\nFinally, the universe might be "open," or have negative curvature. Such universes are sort of saddle-shaped. They are also infinite and unbounded. Parellel lines eventually diverge, and triangles have less than 180 degrees. Open universes expand forever, with the expansion rate never approaching zero.\n\nWhat determines the shape of the universe is its density (and the Cosmological Constant, a sort of anti-gravity force allowed by General Relativity). It is difficult to figure out what the density of the universe actually is, but it seems that the universe is probably flat.
967530
May be a Sphere/Circle of defined polygons or Trees.... with infinite edges and finite branching points....
608577
Depends on the shape of the object being measured.
192822
something to do with the curved space-time continuum
10290
The universe is created by a thought consciousness which manifests in physical reality through a geometric blueprint that we call Sacred Geometry which repeats in cycle giving the illusion of linear time.\n\n12=1+2=3=third dimension - manifestation in 3D\n12 spiraling cones around 1 source = 13 = 4=4th dimension = time\nForming grid programs of eXperience - MatriX - Tricks - Trickster - Duality\nThe boX = 'X' or the hourglass='Z'\n'X' - As is above, So is Below - Emerald Tablets of Thoth=Hermes Trismegistus\nHermes The Magician - Magic - Tricks - Magi - Christ Consciousness Grid\nThoth the ancient Egyptian Scribe who wrote our program\nEmerald = Alchemy - Emerald City = OZ - The WiZard\nAlchemy Wheel of Consciousness based on the spiraling patterns of\nSacred Geometry = SG = StarGate = Wheel of Karma = Alchemy Wheel\nwhich spirals consciousness back to Source\nand is linked with the center - Eye of the Milky Way Galaxy - the Heart
914601
http://mathworld.wolfram.com/PoincareConjecture.html
266432
This is tricky without a particular formula to derive. The master of this was Richard Feynmann. Try his Six Easy Pieces. His lectures on the motion of the planets and his autobiographies are fun easy reading.\n"Surely You're Joking Mr Feynmann"  is the first.\nThe Physics Lecture series are classic physics books.