Find the volume enclosed by the following regions: ; ; .
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