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}]
>
> 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.

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

```

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