Calculating the integral of a uniform Disk?

Questions
,
1

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.

8469Capture.PNG
8469Capture.PNG725Ă—205 15.6 KB

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])
2

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}

3

Thank you for pointing out the error.

4

There are some extra notes to understand the problem evaluation…

8799Capture.PNG
8799Capture.PNG841Ă—473 45.3 KB

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

5

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}]]
6

Thank you Sir for reconsidering the problem.

7

Calculating the integral of a uniform Disk?