Re: help with double integration
- To: mathgroup at smc.vnet.net
- Subject: [mg121779] Re: help with double integration
- From: "Dr. Wolfgang Hintze" <weh at snafu.de>
- Date: Sat, 1 Oct 2011 03:09:37 -0400 (EDT)
- Delivered-to: l-mathgroup@mail-archive0.wolfram.com
- References: <j63t52$6dv$1@smc.vnet.net>
"Salman Durrani" <dsalman96 at yahoo.com> schrieb im Newsbeitrag news:j63t52$6dv$1 at smc.vnet.net... > Hello > > I am trying to use mathematica to do the following double > integration: > > Integrate[ > Integrate[ > r^2*ArcCos[x/r] - x*Sqrt[r^2 - x^2], {x, Sqrt[3] y, > Sqrt[r^2 - y^2]}], {y, 0, r/2}] > > The correct answer is: > r^4(pi^2/144 + pi/(24Sqrt{3}) - 1/32) > > However I am unable to get mathematica to produce the correct result. > I have tried splitting into two integrations and using the > assumptions option (suggested by newsgroup): > > Integrate[ > r^2*ArcCos[x/r] - x*Sqrt[r^2 - x^2], {x, Sqrt[3] y, > Sqrt[r^2 - y^2]}, Assumptions -> {y > 0, r > 2 y}] // Simplify > > The above produces the result for the inner integration but then > > Integrate[ > y^3/3 + 2/3 r^2 Sqrt[r^2 - 3 y^2] + y^2 Sqrt[r^2 - 3 y^2] - > r^2 y (1 + Sqrt[3] ArcCos[(Sqrt[3] y)/r]) + > r^2 Sqrt[r^2 - y^2] ArcSec[r/Sqrt[r^2 - y^2]], {y, 0, > r/2}] // Simplify > > does not produce any result. > > Am I missing something fundamental here. Any help would be > appreciated. > > Thanks > > Kahless > Hello, sometimes Mathematica needs some manual help. Then it works fine as we shall see. Let us start with your last integral which Mathemacia could not solve directly, and let's split in into three parts 1) Integrate[ y^3/3 + 2/3 r^2 Sqrt[r^2 - 3 y^2] + y^2 Sqrt[r^2 - 3 y^2] , {y, 0, r/2},Assumptions->r>0] ((9 + 4*Sqrt[3]*Pi)*r^4)/96 no problem! 2) - Integrate[ r^2 y (1 + Sqrt[3] ArcCos[(Sqrt[3] y)/r]) , {y, 0, r/2}] // Simplify // InputForm -((9 + 7*Sqrt[3]*Pi)*r^4)/144 no problem! 3) Integrate[ r^2 Sqrt[r^2 - y^2] ArcSec[r/Sqrt[r^2 - y^2]], {y, 0, r/2}] // Simplify Integrate[r^2*Sqrt[r^2 - y^2]* ArcSec[r/Sqrt[r^2 - y^2]], {y, 0, r/2}] Problem! In order to solve 3) we change the variable of integration from y to y/r (notice the extra factor r in front of the expression. This takes care of the change in dy) 3.1) Integrate[ r (r^2 Sqrt[r^2 - y^2] ArcSec[r/Sqrt[r^2 - y^2]]) /. y -> r t, {t, 0, 1/2}] // Simplify Integrate[r^2*Sqrt[-(r^2*(-1 + t^2))]* ArcSec[r/Sqrt[-(r^2*(-1 + t^2))]], {t, 0, 1/2}] The problem persists! Now tell Mathematica that r>0 3.2) Integrate[ r (r^2 Sqrt[r^2 - y^2] ArcSec[r/Sqrt[r^2 - y^2]]) /. y -> r t, {t, 0, 1/2}, Assumptions -> r > 0] // Simplify ((-9 + 3*Sqrt[3]*Pi + Pi^2)*r^3)/144 Hurray! Now adding the three terms gives the desired result. 3.3) Unfortunately, simply imposing r>0 in 3) is not sufficient. Hope this helps. Regards, Wolfgang