Once again we use cylindrical coordinates:
For the cone we have, in cylindrical coordinates: r = z,
For the sphere we have: r =
Note that the cone is inverted, so the volume makes an ice-cream cone shape. Where exactly does the cone run into the sphere?
Thats when z = , or z = . This is not quite true since, as it is defined, the cone doesn't extend into the negative z range. So we stick with positive .
This time we break it up into two integrals. We have to change over from the cone to the sphere when we reach z =
Note how in the first integral we integrate r from 0 to z, corresponding to the cone. In the second we integrate corresponding to the sphere.
I got for the volume.