Problem with integrating to find volume

Question

I want to calculate the volume of the hyperboloid between the two planes:

$hyp:{x}^{2}+1/4\,{y}^{2}-{z}^{2}=5$

$plane1:=z+x/2+3=0$

$plane2:=z+x/2-3=0$

Because of symmetry, I confine my bounds to the positive z-axis, and positive y-axis. Now, I'd like to get the intersection between the green plane(plane2) and the hyperboloid, for y-values > 0. Initially i would have solved plane2 and hyp for z, let's say zPlane2 and zHyp, and solve(zHyp=zPlane2), but this doesn't work either. I guess because the information of the y-variable gets lost when solving plane2 for z... Can someone push me in the right direction?

Thanks.

Answer 1

Okay, i got it.

hyp:=x^2+(y/2)^2-z^2=5:
vlak1:=z+1/2*x+3=0:
vlak2:=z+1/2*x-3=0:

I defined them explicit this way:

f(x,y):=solve(vlak1,z);
g(x,y):=solve(vlak2,z);
h(x,y):=solve(hyp,z)[1];

So now it was possible to solve them, to obtain the intersection

solve(g(x,y)=h(x,y),y);