Integration probelm
- To: mathgroup at smc.vnet.net
- Subject: [mg122038] Integration probelm
- From: Jing <jing.guo89 at yahoo.com>
- Date: Mon, 10 Oct 2011 04:27:31 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
Hi, I am trying to use mathmatica to integrate a pretty complicated expression. But there is some problem for the integration. Seeking help... My aim is to integrate the following expression: Integrate[-1/2 y (x + Sqrt[r^2 - y^2]) + 1/4 (-Sqrt[3] x + y) (1/2 (x + Sqrt[3] y) + Sqrt[ r^2 - 1/4 (-Sqrt[3] x + y)^2]) + 1/2 r^2 (-(2 pi)/3 + ArcCos[-y/r] + ArcCos[(-Sqrt[3] x + y)/(2 r)]), {x, -Sqrt[r^2 - y^2], y/Sqrt[3] }, {y, -r*Sqrt[3]/2, 0}] I divide the expression in to 5 parts. 1. Integrate[-1/2 y (x + Sqrt[r^2 - y^2]), {x, -Sqrt[r^2 - y^2], y/Sqrt[3] }, Assumptions -> {y < 0 && r > 0 && y^2 < r^2},{y, -r*Sqrt[3]/2, 0}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] // FullSimplify The answer is -1/144 (-9 + Sqrt[3] \[Pi]) r^4 no problem 2. Integrate[ 1/8 (-Sqrt[3] x + y) (x + Sqrt[3] y) , {x, -Sqrt[r^2 - y^2], y/Sqrt[3] }, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3},{y, -r*Sqrt[3]/2, 0}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] ] The answer is -(3 r^4)/64 no problem 3. First I integrate: 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] }, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] // FullSimplify output is 1/72 (8 Sqrt[3] r^3 - Sqrt[3] r^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - 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]))] Then I integrate : Integrate[ 1/72 (8 Sqrt[3] r^3 - Sqrt[3] r^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - 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]))]), {y, -r*Sqrt[3]/2, 0}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] This time, the Mathmatic can not do integrate, it show me nothing! 4. Integrate[-1/2 r^2 (2 \[Pi] )/3+1/2 r^2*ArcCos[-y/r], {x, -Sqrt[r^2 - y^2], y/Sqrt[3] }, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3},, {y, -r*Sqrt[3]/2, 0}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4*y^2/3}] The answer is -1/72 (-9 + Sqrt[3] \[Pi] + 2 \[Pi]^2) r^4 no probelm. 5. Again, I divide it into two parts. First I integrate: 1/2 r^2 * ArcCos[(-Sqrt[3] x + y)/(2 r) ( it should alter the variable x to t= (-Sqrt[3] x + y)/(2 r). in order to simplify the integration :Integrate[ r^3 ArcCos[t]/Sqrt[3], {t, 0, (Sqrt[3 (r^2 - y^2)] + y)/(2 r)}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] // FullSimplify) Answer is 1/(2 Sqrt[3])r^2 (2 r - Sqrt[ r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])] + (y + Sqrt[3] Sqrt[r^2 - y^2]) ArcSec[(2 r)/( y + Sqrt[3] Sqrt[r^2 - y^2])]) Then, problem occurs again : Integrate[ 1/(2 Sqrt[3]) r^2 (2 r - Sqrt[ r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])] + (y + Sqrt[3] Sqrt[r^2 - y^2]) ArcSec[(2 r)/( y + Sqrt[3] Sqrt[r^2 - y^2])]), {y, -r*Sqrt[3]/2, 0}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] It can not work. and no answer. Based on what I have done, I find that the Mathmatic can not integrate the following expression: 1. Integrate[ 1/(2 Sqrt[3]) r^2 (-Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - y^2])]), {y, -r* Sqrt[3]/2, 0}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] 2. Integrate[ (y + Sqrt[3] Sqrt[r^2 - y^2]) ArcSec[(2 r)/( y + Sqrt[3] Sqrt[r^2 - y^2])], {y, -r*Sqrt[3]/2, 0}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] 3.Integrate[ 1/72 (8 Sqrt[3] r^3 - Sqrt[3] r^2 Sqrt[r^2 + 2 y (y - Sqrt[3] Sqrt[r^2 - 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]))]), {y, -r*Sqrt[3]/2, 0}, Assumptions -> {y < 0 && r > 0 && y^2 < r^2 && r^2 < 4 y^2/3}] When I try to integrate the above equations, the Mathmatic doesn't work. I hope someone can help me to figure it out. Cheers
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