# Volume of region bounded by cone and parabloid

I dont know if anyone will be able to help me, im really stuck on this question!

"Show that the volume of the region bounded by the cone
z=sqrt((x*x)+(y*y)) and the parabloid z=(x*x)+(y*y) is
PI/6"

The bits in the brackets (ie x*x and y*y) are x squared and y squared respectively and sqrt is square root.

Any help would be very much appreciated!

Cheers,

$$\int \sqrt{(x^2 + y^2)} - (x^2 + y^2) dx =~30\deg$$

Originally posted by PrudensOptimus
$$\int \sqrt{(x^2 + y^2)} - (x^2 + y^2) dx =~30\deg$$
30 degrees?

Pi/6 is not 30 degrees. Pi/6 radians is 30 degrees. And you set up the integral wrong. There's more than one variable.

Those two guys intersect at z=1, directly above the circle (on the x-y plane) x2 + y2 = 1, and at the origin.
Within that region, (i.e. inside the cylinder x2 + y2 = 1) the surface of the cone lies above the surface of the paraboloid, so you want the volume bounded by the cone, the cylinder, and the plane z=0
MINUS the volume bounded by the cylinder, the paraboloid, and the plane z=0.

Put your two equations into polar coordinate form & you'll have two very simple double integrals that will give you the result you're looking for.

HallsofIvy