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Calculating the integral of a uniform Disk?

Posted 16 days ago
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I am integrating the following integral as mentioned which is unknown, I am getting somehow incorrect answer. I need it in the form of a elliptic functions. Apparently my approach seems to be right but its actually not correct. Kindly help me with it. enter image description here

Uniform Disk integration:

In[1]:= S = \[Pi] (Rs - \[Beta] \[ScriptCapitalD]LS ) + \[Integral]ArcCos[(-Rs^2 + \
\[Beta]^2 \[ScriptCapitalD]LS^2 + R^2)/(
     2 \[Beta] \[ScriptCapitalD]LS R)] \[DifferentialD]R

Out[1]= \[Pi] (Rs - \[ScriptCapitalD]LS \[Beta]) + 
 R ArcCos[(R^2 - Rs^2 + \[ScriptCapitalD]LS^2 \[Beta]^2)/(
   2 R \[ScriptCapitalD]LS \[Beta])] + (
 I (Rs + \[ScriptCapitalD]LS \[Beta]) Sqrt[
  1 - R^2/(Rs - \[ScriptCapitalD]LS \[Beta])^2] Sqrt[
  1 - R^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2] ((Rs + \[ScriptCapitalD]LS \
\[Beta]) EllipticE[
      I ArcSinh[
        R Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \[Beta])^2)]], (Rs - \
\[ScriptCapitalD]LS \[Beta])^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2] - 
    2 Rs EllipticF[
      I ArcSinh[
        R Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \[Beta])^2)]], (Rs - \
\[ScriptCapitalD]LS \[Beta])^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2]))/(
 R \[ScriptCapitalD]LS \[Beta] Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \[Beta])^2)]
   Sqrt[-((R^4 + (Rs^2 - \[ScriptCapitalD]LS^2 \[Beta]^2)^2 - 
    2 R^2 (Rs^2 + \[ScriptCapitalD]LS^2 \[Beta]^2))/(
   R^2 \[ScriptCapitalD]LS^2 \[Beta]^2))])

In[2]:= M1 = S /. {R -> Rs - \[Beta] \[ScriptCapitalD]LS} 

Out[2]= \[Pi] (Rs - \[ScriptCapitalD]LS \[Beta]) + (Rs - \[ScriptCapitalD]LS \[Beta]) \
ArcCos[(-Rs^2 + \[ScriptCapitalD]LS^2 \[Beta]^2 + (Rs - \[ScriptCapitalD]LS \
\[Beta])^2)/(2 \[ScriptCapitalD]LS \[Beta] (Rs - \[ScriptCapitalD]LS \[Beta]))
   ]

In[3]:= M2 = S /. {R -> Rs + \[Beta] \[ScriptCapitalD]LS} 

Out[3]= \[Pi] (Rs - \[ScriptCapitalD]LS \[Beta]) + (Rs + \[ScriptCapitalD]LS \[Beta]) \
ArcCos[(-Rs^2 + \[ScriptCapitalD]LS^2 \[Beta]^2 + (Rs + \[ScriptCapitalD]LS \
\[Beta])^2)/(2 \[ScriptCapitalD]LS \[Beta] (Rs + \[ScriptCapitalD]LS \[Beta]))
   ]

In[4]:= FullSimplify[M2 - M1]

Out[4]= \[Pi] (-Rs + \[ScriptCapitalD]LS \[Beta])
6 Replies

It seems that you have met an incorrect algebraic simplification. The extrema of your integral are singular points for the primitive, because there is a fraction that evaluates to 0/0, but somehow Mathematica evaluates the fraction to 0:

In[114]:= 
a = (I (Rs + \[ScriptCapitalD]LS \[Beta]) Sqrt[
     1 - R^2/(Rs - \[ScriptCapitalD]LS \[Beta])^2] Sqrt[
     1 - R^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2] ((Rs + \
\[ScriptCapitalD]LS \[Beta]) EllipticE[
        I ArcSinh[
          R Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \[Beta])^2)]], (Rs - \
\[ScriptCapitalD]LS \[Beta])^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2] \
- 2 Rs EllipticF[
        I ArcSinh[
          R Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \[Beta])^2)]], (Rs - \
\[ScriptCapitalD]LS \[Beta])^2/(Rs + \[ScriptCapitalD]LS \[Beta])^2]));
b = (R \[ScriptCapitalD]LS \[Beta] Sqrt[-(1/(Rs - \[ScriptCapitalD]LS \
\[Beta])^2)] Sqrt[-((R^4 + (Rs^2 - \[ScriptCapitalD]LS^2 \[Beta]^2)^2 \
- 2 R^2 (Rs^2 + \[ScriptCapitalD]LS^2 \[Beta]^2))/(R^2 \
\[ScriptCapitalD]LS^2 \[Beta]^2))]);
{a, b, a/b} /. R -> Rs - \[Beta] \[ScriptCapitalD]LS;
Simplify[%]

Out[117]= {0, 0, 0}

You must not simply replace R with Rs - \[Beta] \[ScriptCapitalD]LS, but you have to calculate a limit. Alternatively, do a change of variables and a definite integral:

symbolicResult = 
 Integrate[
  ArcCos[x] D[
    R /. Solve[(R^2 - Rs^2 + \[ScriptCapitalD]LS^2 \[Beta]^2)/(
        2 R \[ScriptCapitalD]LS \[Beta]) == x, R][[2]], x], {x, -1, 
   1}, Assumptions -> 
   Rs + \[Beta] \[ScriptCapitalD]L > 
    Rs - \[Beta] \[ScriptCapitalD]LS > 0]

I checked that the result is correct in the case {\[Beta] = 1, \[ScriptCapitalD]LS = 1, Rs = 2}

Posted 16 days ago

Thank you for pointing out the error.

Posted 16 days ago

There are some extra notes to understand the problem evaluation... enter image description here

\beta is the image angle formed as a result of image formed due to reflection of light rays of the sun.

You can see from the plot that the primitive is discontinuous at the point where you evaluate it:

Block[{\[Beta] = 1, \[ScriptCapitalD]LS = 1, Rs = 2},
 ReImPlot[S,
  {R, Rs - \[Beta] \[ScriptCapitalD]LS - 1, 
   Rs + \[Beta] \[ScriptCapitalD]LS + 1}]]
Posted 16 days ago

Thank you Sir for reconsidering the problem.

Posted 15 days ago

Calculating the integral of a uniform Disk?

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