How to calculate this volume?
Using the parametric equivalents
$$C:=\lbrace (\cos(\theta_1)+1,\sin(\theta_1),0)\rbrace$$
$$C':=\lbrace (\cos(\theta_2)-1,0,\sin(\theta_2))\rbrace$$
therefore
$$\overline{C}:=\lbrace(t\cos(\theta_1)+t+(1-t)\cos(\theta_2)-(1-t),t\sin(\theta_1),(1-t)\sin(\theta_2))\rbrace$$
Can you take it from there?
OK guys - try this:
Let
$$y=ty', y' \in [-1,1]$$
$$z=(1-t)z', z'\in [-1,1]$$
Therefore
$$x=t(\cos(\pm \arcsin(y'))+1)+(1-t)(\cos(\pm\arcsin(z'))-1)$$