# Volume of paraboloid that is cut with plane

How to calculate the volume of the paraboloid :

x^2 + y^2 = z

that is cut with x + y + z = 5 plane.

Please give several methods if you can. Thank you very much for answers.

• I have tried projecting the ellipse on the xy plane as circle with center that is a little bit off from (0;0) point. But I'm not sure if thats correct. I have integrated that ,but I'm not sure if it is correct answer and if that ellipse projects really as a circle on the xy plane. Another idea I have is to use Mean value theorem to find the mean point. Because D1(upper section of paraboloid that has more volume) and D2(lower section of paraboloid that has lower volume) they are different. And then integrate x^2 + y^2 = z simply having plane cutting paraboloid 180* degrees. Thats my ideas. – Tom May 1 '16 at 12:44
• @Tom You gave a lot of good insights into your thinking process in your comment; even better would be to edit the question itself to add all of that information. – David K May 1 '16 at 13:30