# Internal theoretical figure to a physical idea of

Internal Assessment                        Carson Graham SecondaryMs. J. Dai2017****all texts with “>>text<<" will be removed prior to final submissionProof of Gabriel's Horn (Torricelli's trumpet)Does the Mathematical paradox of Gabriel's Horn affect its existence?ContentsIntroduction    >>personal engagement<>explain math theory<>avoid mistakes,apply concept, include graphs, tables, all steps<>comment and briefly reflect on results make sure to interpret results well<>touch on how original research question wasn’t going to work<<{equation number}     -     Reference number for equationsFigure x.y        -     Reference number for graphs, data, and other figures1)  Introduction    Gabriel's Horn or Torricelli's Trumpet is a unique geometric figure due to its interesting properties. It is known for being discovered and studied by Evangelista Torricelli(15 October 1608 – 25 October 1647), an Italian physicist and mathematician renowned for his creation of the mercury barometer and Torricelli's equation(Britannica, 2017). This figure has been titled alluding to the Archangel Gabriel's Horn he blows on Judgement Day, symbolizing the connection between finite and infinite(Theindeliblelifeofme, 2014). The property that causes it to differ from others, is that it has infinite surface area, but has finite volume.

Many have disputed(C. Wijeratne, 2015), claiming that there are paradoxes with the concept of  Gabriel’s Horn and cannot possibly exist.I have always been intrigued by different topics of debate, and Gabriel’s Horn has especially peaked my interest due to its paradoxical nature.

Personally, I did not believe a shape of finite volume and infinite surface area could possibly exist. My mistake here was relating a theoretical figure to a physical idea of possible figures. However this did not sway me and this topic is worth exploring because Therefore my goal for this research is to discover how a mathematically paradoxical figure can exist, as well as possible applications for Gabriel’s Horn, to find possible uses for the convenience of other maths, similar to the function of complex numbers such as i. I will obtain the data for this research by looking and using multiple past research stemming from Torricelli’s investigations majorly second hand sources because Torricelli’s work was done in italian, although some formulae may be evident I would not be able to fully understand the contexts. Furthermore, I will derive the volume and surface area myself, to understand how this mathematically and seemingly logically defying shape can exist.

2) The figure of Gabriel’s HornGeometry   >>more explanation/detail<<    Gabriel's Horn is a solid of revolution that is created using the following properties. Figure 2.1 First, the function of y=1xis given. Figure 2.

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2 Second, the domain is limited to x1 (area is represented in red) Figure 2.3 Third, function is revolved around the x axis when 1    x +,resulting in a cone shape, which then forms the shape of a funnel        >>Change x=1 in figure to y=1<

4 The volume of the disks are r2hwhich is the formula for calculation of a cylinder. (represented in yellow)>>Image citationhttp://www.worldcat.org/title/single-variable-calculus-early-transcendentals/oclc/52203359?page=citation <

Similarly, surface area should also be infinite from an endless endpoint, but with the knowledge that volume is indeed not infinite, the surface area could be anything. Therefore, I looked to prove or find evidence if the surface area was infinite or otherwise.3) Calculations    Variables, formulae and terminology clarifications        {1}        Volume of cylinder V r2h                    Where  r= radius                h= height        a integrated to b  ab{2}                Surface Area of a disk(refer to Figure 2.4)  C 2r            Where  r= radius                Centroid meana) VolumeTo start, we have our area formula for 1x from point  x=1integrated to x=.{3}    11xdxNext, we obtain V = volume of a solid of revolution by using the general formula found within the math HL syllabus.                    {4}    V=h0x=ax=bfx2h            (Since the volume of a cylinder is  r2h,                            we use the function of x for r, where r= radius of cylinder)We then convert the 2 dimensional area formula for 1xinto the general formula for the volume of a  solid of revolution    {5}    V=h0x=1x=1x2h            Which simplifies to{6}    V=11x2dx                Then, {7}    V=a1a1x2dx            We convert 1 to a1a for calculation purposes, as is not a number        For fx=1×2 , We use the power rule.

..{8}            xndx=xn+1n+1            Where n = -2…to find the integral Fx= x-1-1 = -1x {9}      V=a-1×1 a                We use abecause we can’t use directly as is not a numberAnd we continue by substitute x for 1 and a, V=a-1a–11       =a-1a+1   =-0+1                Since -1a converges to 0 as a   =1    {10}    V=?The volume of Gabriel’s Horn is .

From the calculations above, I have concluded that the volume of Gabriel’s Horn isunits3, which contradicted my original hypothesis that a solid with infinite distance between a and b. I was surprised at this fact initially, but after some thought on the process of the creation of Gabriel’s Horn, I realized that it would be flawed to compare a theoretical figure to physically possible shapes. b) Surface AreaIn order to find the surface area of Gabriel’s Horn we must first derive a formula for the surface area of a solid of revolution because this knowledge is not found within the math HL syllabus. We first use the formula for the circumference of the disks {reference equation 2}    C=2r                    Where r = radiusWe then substitute our function 1xfor r{11}    C=21xPappus’ Centroid 1st Theorem states that     Sa=?d                Where     Sa= Surface Area                         ?= Arc Length                        d=Distance travelled by the solid of revolutions       geometric centroidA centroid is defined as the geometric center or arithmetic mean position of all points in a shape(Bourke, 1988)>>derived surface area of solid of revolution using Pappus’s centroid theorem, the Pythagorean theorem and arc length formula already, have not typed<1    {19}    1= 1{20}    ? 1+x> 1                Adding any positive value x will cause it to be greater than 1 because x4is always positive{21}    ? 1+1×4>1{22}    ? 1×1+1×4>1×1                Adding 1xto both sides of the inequality{23}    ? 11×1+1x4dx>11x1dx            Adding the improper integral 1 to both sides of the inequality{24}    ?211×1+1x4dx>211x1dx        Adding 2to both sides of the inequalityQuestion #9 from section 22E.

2 of the HL coursework(D.Martin, 2012) states that “the ..

. area from x=1 to infinity is infinite for the function y=1x.” This is shown as {25}    A=11xdx = If this were to be proven true, then     211x1dx=  would be true as well because 11xdx < 211x1dxTherefore, we need to prove that  A=11xdx is infinite{26}    A=11xdx                    =b1b1xdx                    We use bbecause we can't use directly as is not a number{27}    =blnx|1b                Because for fx=1x   ,   f'x=lnx  {28}    =b(lnb-ln1)            {29}    =blnb                    ln1 is irrelevant if b{30}    A=Since 11xdx  <  211x1dx  < 211x1+1x4dx  ,        {reference equation 24, 26}    < 211x1+1x4dx?211x1+1x4dxmust also be ? Surface Area of Gabriel's Horn is 4)  Results Analysis    The results I got from calculating Gabriel's Horn were fascinating, due to the fact that I had no idea that this solid was possible, and it seemed to contradict the basic common sense that if something is infinitely long, it must hold infinite volume.

Furthermore, as previously stated, in the course work of the math HL work within the textbook(D.Martin, 2012), I noticed that it stated that “We call thisthe relationship between the function of 1x where x1and Gabriel’s horn a mathematical paradox.” This statement is correct, proven in {reference equation 30}    Proving the relationship of 1x where x1and Gabriel’s horn is indeed mathematically paradoxical, however the fact that states the Gabriel’s Horn does is exist is not paradoxical, concluded from calculations and evidence gathered in section 3) Calculations.5)  Conclusion>>unfinished<