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still struggling about integration

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122337] still struggling about integration
  • From: Jing <jing.guo89 at yahoo.com>
  • Date: Tue, 25 Oct 2011 06:19:43 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

Hi,

I want to integration two euqations:

1/4 (-Sqrt[3] x + y) Sqrt[r^2 - 1/4 (-Sqrt[3] x + y)^2];

and -r^2 ArcCos[(-Sqrt[3]x+y)/(2r)]/2

For both of them, x is integrated from -Sqrt[r^2-y^2] to -y/Sqrt[3]-s; y is from (-Sqrt[3]s-Sqrt[12r^2-9s^2])/4 to (-Sqrt[3]s+Sqrt[12r^2-9s^2])/4. 
s, r are two constant and s>0 and Sqrt[3]s/2<r<s.

I used to ask quite same questions, some one gives me hints. But I find this time it can not work.


For example, for the first equation:
m18 = Integrate[(a Sqrt[r^2 - a^2])/Sqrt[
    3], {a, (2 y + Sqrt[3] s)/2, (Sqrt[3 (r^2 - y^2)] + y)/2 }, 
   Assumptions -> {y < 0 && r > 0 && r^2 > 4 y^2/3 && s > 0 && 
      Sqrt[3] s/2 < r < s && a^2 < r^2 && a > 0}] // FullSimplify

Result:1/(24 Sqrt[
  3]) ((4 r^2 - 3 s^2 - 4 Sqrt[3] s y - 4 y^2)^(3/2) - 
   r^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[(r - y) (r + y)])] - 
   2 y^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[(r - y) (r + y)])] + 
   2 Sqrt[3]
     y Sqrt[(r - y) (r + y) (r^2 + 
       2 y (y - Sqrt[3] Sqrt[(r - y) (r + y)]))])

someone suggest to me that I can use y=-r*sin(t), this variable alteration and then do the integration. I folowed this time, 

exp1 = Simplify[m18 /. {y -> -r Sin[j]}, r > 0 && 0 < j < Pi/3]
result:
1/(24 Sqrt[
  3]) ((2 r^2 - 3 s^2 + 2 r^2 Cos[2 j] + 4 Sqrt[3] r s Sin[j])^(
   3/2) - 2 r^3 Sqrt[2 - Cos[2 j] + Sqrt[3] Sin[2 j]] + 
   r^3 Cos[2 j] Sqrt[2 - Cos[2 j] + Sqrt[3] Sin[2 j]] - 
   r^3 Sin[2 j] Sqrt[6 - 3 Cos[2 j] + 3 Sqrt[3] Sin[2 j]])

Then I do the integartion for j.

Integrate[-r Cos[j] exp1, {j, ArcSin[(-Sqrt[3] s - Sqrt[12 r^2 - 9 s^2])/
    4r], ArcSin[(-Sqrt[3] s + Sqrt[12 r^2 - 9 s^2])/4r]}, 
  Assumptions -> {r > 0 && r^2 > 4 y^2/3 && s > 0 && 
     Sqrt[3] s/2 < r < s}] // FullSimplify

But it takes a long time to run and finally show"No more memory available.
Mathematica kernel has shut down." On the screen.

For the second equation, same thing happens. I can do the integration for x, but for y, I can not.

PS: I hope the result is real number and "no imaginary part".
Can someone help me to solve this integration problems.

Thanks.



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