integration problem

*To*: mathgroup at smc.vnet.net*Subject*: [mg122236] integration problem*From*: Jing <jing.guo89 at yahoo.com>*Date*: Sat, 22 Oct 2011 06:05:16 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com

Hi, I want to integration two eqations: 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. For example, for the first equation: 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: 1/72 (-3 Sqrt[3] s^2 Sqrt[4 r^2 - 3 s^2 + 4 Sqrt[3] s y - 4 y^2] + 12 s y Sqrt[4 r^2 - 3 s^2 + 4 Sqrt[3] s y - 4 y^2] - 4 Sqrt[3] y^2 Sqrt[4 r^2 - 3 s^2 + 4 Sqrt[3] s y - 4 y^2] - 2 Sqrt[3] y^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])] + 6 y Sqrt[(r^2 - y^2) (r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2]))] + Sqrt[3] r^2 (4 Sqrt[4 r^2 - 3 s^2 + 4 Sqrt[3] s y - 4 y^2] - Sqrt[ r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])])) I used to ask a question quite same, 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: 1/72 (2 r^2 Sqrt[ 6 r^2 - 9 s^2 + 6 r^2 Cos[2 j] - 12 Sqrt[3] r s Sin[j]] - 3 s^2 Sqrt[ 6 r^2 - 9 s^2 + 6 r^2 Cos[2 j] - 12 Sqrt[3] r s Sin[j]] - 12 r s Sin[j] Sqrt[ 2 r^2 - 3 s^2 + 2 r^2 Cos[2 j] - 4 Sqrt[3] r s Sin[j]] - 3 r^3 Sin[2 j] Sqrt[2 - Cos[2 j] + Sqrt[3] Sin[2 j]] - 2 r^3 Sqrt[6 - 3 Cos[2 j] + 3 Sqrt[3] Sin[2 j]] + Sqrt[3] r^2 Cos[ 2 j] (2 Sqrt[ 2 r^2 - 3 s^2 + 2 r^2 Cos[2 j] - 4 Sqrt[3] r s Sin[j]] + r Sqrt[2 - Cos[2 j] + Sqrt[3] Sin[2 j]])) Then I hope to do the integration and subsitute the two limits for y( if I directly put the two bounds of y into the integration, it will take over half hour to run and finally, it will show "no memroy"]: m19 = Integrate[-r Cos[j] exp1, j] It gives me really long euqtion, about hundreds lines long. For the second equation, the result is really long, it is about hundreds line long and there is I inside the equation. I don't know why "I" this symbol will appear in the equation. Can someone help me to solve this integration problems. Thanks. Jing