Google "Steinmetz solid" for information and proofs.
You are given three circular cylinders, each of which is located along its coordinate axis. The axis of the first cylinder coincides with the axis ; second cylinder axis coincides with the axis ; third cylinder axis coincides with the axis . We have a symmetry, so we should calculate the amount of body parts in the first oktant, and the answer multiplied by . Next, we consider the case . Then the upper limit of integration for we can determine from the equation which gives us a lower value for , other than equation . The field of integration in the plane , we can determine from the equation , i.e. this area is 1/8 of unit circle.
So we have , .
Next we will use cylindrical coordinates to calculate the integral