Find the volume enclosed by the following regions:

x^2 + y^2 = 1;

y^2 + z^2 = 1;

z^2 + x^2 = 1:

Printable View

- Feb 4th 2009, 07:18 PMAndreametmultiple integrals
Find the volume enclosed by the following regions:

x^2 + y^2 = 1;

y^2 + z^2 = 1;

z^2 + x^2 = 1: - Feb 5th 2009, 10:18 AMOpalg
Google "Steinmetz solid" for information and proofs.

- Feb 5th 2009, 08:50 PMDeMath
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

- Feb 7th 2009, 01:45 PMDeMath
Look this graphic interpretation

http://s47.radikal.ru/i118/0902/16/ab60d224a366.jpg