Calculating the Volume of a Region Bounded by Hyperboloid Cylinders

[siddharth]

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

The Volume will be
$$\int \int \int_{V} dx dy dz$$

which I think is,

$$\int \int_{D} (\frac{49}{y}-\frac{25}{y})dx dy$$

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 $\frac{1}{2\sqrt{abc}}$). 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.