definite integralsintegrationinterpolationnumerical methods
I am trying to calculate the surface area of revolution for The Gherkin, an egg-shaped building in London, UK. Not sure about how to obtain the equation of the curve but I have the data points that allowed me to graph it in Excel but the equation, as it turns out, isn't simply quadratic.
Pls does anyone know any graphing programs that can determine unusual curve equations from a list of data points?? Thanks in advance.
I should not use a strict polynomial regression.
Looking at the data $(h_i,\rho_i)$, I first had a look at the plot of $\log \left(\frac{\rho_i}{\rho_1}\right)$ as a function of $h$. This is quite nice and using a model $$\log \left(\frac{\rho}{\rho_1}\right)=\sum_{k=1}^n a_k \left(\frac {h}{10^4}\right)^k$$ I obtained a very good fit $(R^2=0.999977)$ using $n=4$. For this model, the parameters are all significant as shown below $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ a_1 & -0.2457039080 & 0.006144797 & \{-0.25909227,-0.23231555\} \\ a_2 & -0.0351850842 & 0.001530225 & \{-0.03851916,-0.03185101\} \\ a_3 & +0.0021044026 & 0.000110197 & \{+0.00186430,+0.00234450\} \\ a_4 & -0.0000390562 & 0.000002385 & \{-0.00004425,-0.00003386\} \end{array}$$ Using $n>4$ leads to non-significant parameters.
So, the model would be $$\rho=23.77\, \exp \left(\sum_{k=1}^4 a_k \left(\frac {h}{10^4}\right)^k \right)$$
Below are compared the experimental values and the predicted values from this model. $$\left( \begin{array}{ccc} h & \text{measured} & \text{predicted} \\ 0 & 23.77 & 23.770 \\ 5000 & 20.48 & 20.843 \\ 10000 & 17.56 & 17.986 \\ 15000 & 14.96 & 15.296 \\ 20000 & 12.67 & 12.839 \\ 25000 & 10.66 & 10.651 \\ 30000 & 8.91 & 8.743 \\ 35000 & 7.38 & 7.112 \\ 40000 & 5.87 & 5.739 \\ 45000 & 4.62 & 4.600 \\ 50000 & 3.64 & 3.665 \\ 60000 & 2.26 & 2.297 \\ 70000 & 1.39 & 1.423 \\ 80000 & 0.86 & 0.877 \\ 90000 & 0.56 & 0.541 \\ 100000 & 0.33 & 0.335 \\ 150000 & 0.037 & 0.0366 \\ 200000 & 0.0053 & 0.00534 \\ 250000 & 0.00065 & 0.000649 \end{array} \right)$$
As @Minus One-Twelfth pointed out in the comments: this phenomenon is called Gabriel's horn.
Gabriel's horn is a geometric figure which has infinite surface area but finite volume.
How you can interpret the phenomenon:
You can treat the horn as a stack of disk on top of each other with radii that are diminishing. Every disk has a radius $r = \frac{1}{x}$ and an area of $πr^2$ or $\frac{π}{x^2}$ .
The series $\frac{1}{x}$ diverges but $\frac{1}{x^2}$ converges. So the area under the curve is infinite and the volume of the revolution is finite.
This creates a paradox: you could fill the inside of the horn with a fixed volume of paint, but couldn't paint the inside surface of the horn. This paradox can be explained by using a 'mathematically correct paint', meaning that the paint can be spread out infinitely thin. Therefore, a finite volume of paint can paint an infinite surface.
Best Answer
polyfit in Matlab is great. Returns a polynomial of degree n
Something like:
X = [ x co-ordinates in order (with spaces in between) ];
Y = [ y co-ordinates in order (with spaces in between) ];
p = polyfit(X,Y,n)
http://uk.mathworks.com/help/matlab/ref/polyfit.html
Hope it helps