(ℝ^3): Three Dimensional Coordinates

Example 12.1.34

Let's plot the region and verify that we have the correct solution.

RegionPlot3D[ 1≤(x^2)+(y^2)+(z^2)≤4,{x,-2,2},{y,-2,2},{z,-2,2},PlotStyle->Directive[Yellow,Opacity[0.5]],PlotPoints->36]

The mesh makes it a bit hard to see, but you can definitely see that there are two spheres, one inside the other. Let's remove the mesh and only plot for y>0 to get a better view of the situation. The symbol && means  "and".

RegionPlot3D[1≤(x^2)+(y^2)+(z^2)≤4&&y>0,{x,-2,2},{y,-2,2},{z,-2,2},PlotStyle->{{Yellow,Opacity[0.5]},{Blue}},Mesh->None,PlotPoints->36]

That's Pretty Sweet!

Example 12.1.52

Let's plot this to see if we have the right solution. Rotate the plot to see  where the sphere is centered. Looks like we got it!

ContourPlot3D[3x^2+3y^2+3z^2+2y-2z==0,{x,-1,1},{y,-1,1},{z,-1,1},AxesLabel->{"x","y","z"}]