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[1]:= Integrate[r, {z, 0, (1 - r^2)^(1/2)}]
>
> 2
>Out[1]= r Sqrt[1 - r ]
>
>In[2]:= Integrate[%, {r, Sin[t], 1}]
>
> 2 2
> Cos[t] Sqrt[Cos[t] ]
>Out[2]= ---------------------
> 3
>
>In[3]:= Integrate[%, {t, 0, Pi/2}]
>
> 2
>Out[3]= -(-)
> 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
Copyright 1988-91 Wolfram Research, Inc.
-- NeXT graphics initialized --
In[1]:= Integrate[r,{r,Sin[t],Sqrt[1-z^2]}]
2 2
1 - z Sin[t]
Out[1]= ------ - -------
2 2
In[2]:= Integrate[%,{t,0,ArcCos[z]}]
2
(1 - 2 z ) ArcCos[z] Sin[2 ArcCos[z]]
Out[2]= -------------------- + ----------------
4 8
In[3]:= Integrate[%,{z,0,1}]
2
Out[3]= -
9
Dave Seaman