struggling in integration

• To: mathgroup at smc.vnet.net
• Subject: [mg122362] struggling in integration
• From: Jing <jing.guo89 at yahoo.com>
• Date: Wed, 26 Oct 2011 17:40:14 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com

```Hi,

I want to integration two equations:

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 the same question but the integration area is some kind of different. Some one gave me the hints that I can use variable change when trying to integrate y, y=-r*sint.

But probelm occurs this time.

For example, for the first equation:
First integrate x:
m18 = Integrate[
1/4 (-Sqrt[3] x + y) Sqrt[
r^2 - 1/4 (-Sqrt[3] x + y)^2], {x, -Sqrt[r^2 - y^2], -y/Sqrt[3] -
s }, Assumptions -> {y < 0 && r > 0 && r^2 > 4 y^2/3 && s > 0 &&
Sqrt[3] s/2 < r < s}]

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)]))])

No probelm. Then do the integartion for y and use y=-r*sinj, j is from ArcSin[(-Sqrt[3] s - Sqrt[12 r^2 - 9 s^2])/(-4 r)] to ArcSin[(-Sqrt[3] s + Sqrt[12 r^2 - 9 s^2])/(-4 r)]

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

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

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

Can someone help me to solve this integration problems. And I hope there is nor imaginary number appears in the final equation.

Thanks a lot.

```

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