Study on Spherical Wedge

Here's a start perhaps? Using

h = 4; r = 3; x1 = 1;
Clear[p1, p2]; 
First[NSolve[p1^2 + p2^2 == 6^2 && x1/r == p1/p2, {p1, p2}]] /. Rule -> Set;

we have

pl1 = ParametricPlot3D[3 {Cos[u] Sin[v], Sin[u] Sin[v], Cos[v]},
   {u, π/2, ArcTan[9/3]}, {v, 0, π/2},
   Mesh -> None, Boxed -> False, Axes -> None, 
   PlotStyle -> {Red, Opacity[0.8], Specularity[White, 5]}];
pl2 = Graphics3D[{
    Arrow[{{0, 0, 0}, {0, 0, 5}}], Arrow[{{0, 0, 0}, {0, 5, 0}}], Arrow[{{0, 0, 0}, {5, 0, 0}}],
    Dashed,
    {Red, Line[{{1, 0, 0}, {1, 3, 0}, {0, 3, 0}}]},
    {Blue, Line[{{0, 0, 0}, {0, h, 0}, {0, h, h}, {0, 0, h}, {0, 0, 0}}], Text["X = 0", {0, h, h}, {1.5, 1}]},
    {Brown, Line[{{0, 0, 0}, {p1, p2, 0}, {p1, p2, h}, {0, 0, h}, {0, 0, 0}}], Text["Y = 3x", {p1, p2, h}, {1.5, 1}]},
    {Green, Line[{{0, 0, 0}, {0, h, 0}, {h, h, 0}, {h, 0, 0}, {0, 0, 0}}], Text["Z = 0", {h, h, 0}, {0, -2}]}
    }];
pl3 = ParametricPlot3D[
   r {Cos[ArcTan[9/3]] Sin[t], Sin[ArcTan[9/3]] Sin[t], Cos[t]},
   {r, 0, 3}, {t, 0, π/2},
   Mesh -> None, Boxed -> False, Axes -> None,
   PlotStyle -> {Red, Opacity[0.5], Specularity[White, 5]}];
pl4 = ParametricPlot3D[r {0, 3 Cos[t], 3 Sin[t]},
   {r, 0, 1}, {t, 0, π/2},
   Mesh -> None, Boxed -> False, Axes -> None,
   PlotStyle -> {Red, Opacity[0.5], Specularity[White, 5]}];

Then:

Show[pl1, pl2, pl3, pl4,
 PlotRange -> All, {ViewPoint -> {2, 2.7, 0.6}, ViewVertical -> {0.1, 0.2, 1.}}
 ]

my stab at it. Afraid the quality of the discretization is not so great.

g here is your first Graphics3D

Show[{g, RegionPlot3D[
   RegionIntersection[Ball[{0, 0, 0}, 6], 
    Parallelepiped[{0, 0, 0}, 6 {{0, 1, 0}, {1/3, 1, 0}, {0, 0, 1}}]],
    PlotPoints -> 20, PlotStyle -> Opacity[.25]]}]

this starts to look good with PlotPoints->200 , but still not remotely publication quality.

a similar approach, not as good but maybe of some use.

r = BoundaryDiscretizeRegion[
   RegionIntersection[
    Ball[{0, 0, 0}, 6], 
    Parallelepiped[{0, 0, 0}, 6 {{0, 1, 0}, {1/3, 1, 0}, {0, 0, 1}}]],
    MaxCellMeasure -> .01];
Show[{g, Graphics3D[{Opacity[.25], EdgeForm[None], 
    MeshPrimitives[r, 2]}]}]