I need to calculate the volume of the region bound by :
- The cone $z^2=x^2+y^2$
- The plane $z=2x+2y-2$
- The plane $z=4$
I have already tried setting up a triple integral but I am having some problems determine the upper and lower limits as functions of $y$ and $z$. I already have the values of $z$ ranging from $(2+4\sqrt 2)/7$ and $4$.
What I have so far : $$\int_{(2+4\sqrt 2)/7}^4\,\iint f(x,y,z)\,dx\,dy\,dz$$
And I just about managed to compute the bounds for $z$ by equating $\sqrt{x^2-y^2} = 2x + 2y - 2$, and projecting the resultant hyperbola onto the $xy$–plane to find the vertex.
I'm having trouble determining what the bounds of the inner integrals should be, along with $f(x,y,z)$, what should I do?