This is the 3D contour plot of exactly the author's equations: {x^2 + y^2 + z^2 == 4, x^2 + y^2 - 2 x == 0}
ContourPlot3D[{x^2 + y^2 + z^2 == 4, x^2 + y^2 - 2 x == 0}, {x, -2,
2}, {y, -2, 2}, {z, -2, 2}, AxesOrigin -> {0, 0, 0},
AxesStyle -> Thick]
Which gives the above plot!
To calculate the volume, you have to use solids equivalent to those equations:
Graphics3D[{Ball[{0, 0, 0}, 2], Cylinder[{{1, 0, -2}, {1, 0, 2}}, 1]},
Axes -> True]
Which is different from Simon's plot due to the centre of the sphere not being the author's
The volumes of the ball, the cylinder and their intersection can be perfectly calculated with Mathematica as I did in my first answer:
In[101]:= R1 = Ball[{0, 0, 0}, 2];
R2 = Cylinder[{{1, 0, -2}, {1, 0, 2}}, 1];
R = RegionIntersection[R1, R2];
Volume /@ {R1, R2, R}
Out[104]= {(32 \[Pi])/3, 4 \[Pi], 16/9 (-4 + 3 \[Pi])}