Calculating the Volume

siddharth

The question is, Find the volume of the region bounded by the hyperboloid cylinders
[tex] xy=1, xy=9, xz=36, yz=25, yz=49 [/tex]

The Volume will be
[tex] \int \int \int_{V} dx dy dz [/tex]

which I think is,

[tex] \int \int_{D} (\frac{49}{y}-\frac{25}{y})dx dy [/tex]

The problem I now have is in determining the Domain of Integration D in the x-y plane. I know I am supposed to project the cylinder on to the x-y plane and then find the domain D, but I am stuck. How do I find the curves, y=f(x), which determines this domain?

siddharth

I just thought of something else. I could transform the co-ordinates so that,
xy=a, yz=b and zx=c and then find the Jacobian (which i get as [itex] \frac{1}{2\sqrt{abc}} [/itex]). I think the plane z=0 must also be given as a boundary. Then the shape of the region in the new co-ordinate system will be a cuboid.