There are to pretty similiar volumes, but there is something in the calculation, that I dont understand clearly and I would be glad if someone help me.
The first (Vivani volume): a volume that bounded by a cylinder: $$x^2+y^2=ax\:;\:a>0$$ and by a ball: $$x^2+y^2+z^2=a^2$$ The second one: $$\left(x^2+y^2+z^2\right)^2=z$$ The third: $$\left(x^2+y^2+z^2\right)^3=z^4$$ So in all the solutions, without getting into details of calculations, we calculate the volume of when: $x,y,\:z\ge 0$, but then I dont understand why we multiple by 4 the first and the second and the third by 8. What makes this diffrence ?
To be more accurate,the first by cylindrical substitution becomes:$$4\int _0^{\frac{\pi }{2}}d\theta \int _0^{acos\theta }dr\int _0^{\sqrt{a^2-r^2}}r\:dz\:\:\: $$ The the second by spherical substitution becomes: $$4\int _0^{\frac{\pi }{2}}d\theta \int _0^{\frac{\pi }{2}}d\phi \int _0^{\sqrt[3]{cos\phi \:}}r^2sin\phi \:dr\:\:\:\:$$ And the third: $$8\int _0^{\frac{\pi }{2}}d\theta \int _0^{\frac{\pi }{2}}d\phi \int _0^{cos^2\phi \:}r^2sin\phi \:dr\:\:\:\:$$
P.S: we can understand it much more clearly by using 3D Calc Plotter:[3D Calc Plotter][1] [1]: http://web.monroecc.edu/manila/webfiles/calcNSF/JavaCode/CalcPlot3D.htm
But how do we conclude it from the equations ? As I understand it after I draw the first one (which is the easiest to draw it's graph because it is a moved cylinder inside a ball who's radius is just as the whole diameter of the ball), in the first one it can be concluded easily from the graph of the bounded volume (because by assuming x,y,z > 0, it is exactly a quarter of the whole volume), but how do we see it in the second and the third? Is it enough to see, that the second isn't symmetric to z axe and the third is? And thats why the third by that partition is a 8th part and in the second it is only a quarter?
How will you decide or explain why we multiple by 4 or 8 each integral to calculate the whole volume (in each example)?
You really need to "see the object in your mind" to understand what's going on. It is all about the symmetries of the object. Only the last object is symmetric w.r.t to all sign changes: $x\leftrightarrow -x$, $y\leftrightarrow -y$,$z\leftrightarrow -z$. Geometrically these are reflections w.r.t. the coordinate planes. The first object is not symmetric under the change of sign of $x$, and the second is not symmetric under the change of sign of $z$.
Those reflections generate a group of 8 symmetries, and if the object has all of them it suffices to calculate the volume of the part in the first octant $x,y,z\ge0$. When there is only a 4-fold symmetry we cannot restrict the asymmetric coordinate to have only positive values.