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Re: A negative volume!


"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





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