       Re: A negative volume!

• To: mathgroup at yoda.physics.unc.edu
• Subject: Re: A negative volume!
• From: ags at seaman.cc.purdue.edu (Dave Seaman)
• Date: Thu, 12 Mar 92 11:48:05 EST

```"Roger B. Kirchner" <kirchner at cs.umn.edu> writes:

>Let V be the volume of the solid inside the first octant of the unit
>sphere and outside the cylinder with cylindrical equation r =
Sin[t].
>Computing in cylindrical coordinates,

>
>In:= Integrate[r, {z, 0, (1 - r^2)^(1/2)}]
>
>                    2
>Out= r Sqrt[1 - r ]
>
>In:= Integrate[%, {r, Sin[t], 1}]
>
>              2            2
>        Cos[t]  Sqrt[Cos[t] ]
>Out= ---------------------
>                  3
>
>In:= Integrate[%, {t, 0, Pi/2}]
>
>          2
>Out= -(-)
>          9
>
>Thus V = -2/9!
>
>Anybody have any suggestions on how to avoid this kind of problem?

A change in the order of integration seems to do the trick.

Mathematica 2.0 for NeXT
-- NeXT graphics initialized --

In:= Integrate[r,{r,Sin[t],Sqrt[1-z^2]}]
2         2
1 - z    Sin[t]
Out= ------ - -------
2         2

In:= Integrate[%,{t,0,ArcCos[z]}]
2
(1 - 2 z ) ArcCos[z]   Sin[2 ArcCos[z]]
Out= -------------------- + ----------------
4                    8

In:= Integrate[%,{z,0,1}]
2
Out= -
9

Dave Seaman

```

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