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A negative volume!


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?
Roger Kirchner





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