Double Integrals over General Regions: Examples

1. Find the volume of the solid region bounded by the paraboloid z=4-x^2-2y^2 and the xy plane.

Anwser:

By setting z=0, we find that the region the paraboloid takes up on the xy is a ellipse  4=x^2+2*y^2.  Thus, the region of integration is: x=-2..2, y=-sqrt((4-x^2)/2)..sqrt((4-x^2)/2).

Thus the volume is given by the following integral:

>	int(int(4-x^2-2*y^2,y=-sqrt((4-x^2)/2)..sqrt((4-x^2)/2)),x=-2..2);

2. Find the volume of the solid region R bounded above by the paraboloid z=1-x^2-y^2 and below by the plane z=1-y.

Anwser:

First find the bounds by equating the z values: 1-x^2-y^2=1-y  goes to x^2=y-y^2

Because R is the difference between the volume under the paraboloid and the volume above the plane, you have:

>	int(int(1-x^2-y^2,x=-sqrt(y-y^2)..sqrt(y-y^2)),y=0..1);.