Volume Generated by Hyperboloid & Plane

[semc]

Volume generated[Solved]

Use polar coordinates to find the volume of the given solid enclosed by the hyperboloid -x²-y²+z²=1 and the plane z=2.

I wrote out the integral $$\int\int 1+r^2 dr d\vartheta$$ integrated from 0 to $$2\pi$$ and r integrated from 0 to sqrt3 and volume enclosed gotten to be $$15\pi$$/2. So this is the volume enclosed by the hyperboloid and the xy-plane correct? So I wanted to use the volume of the cyclinder to subtract off the volume found previously. But the volume is $$6\pi$$ so if I subtract i would get negative volume?

 

[jav]

Use polar coordinates to find the volume of the given solid enclosed by the hyperboloid -x²-y²+z²=1 and the plane z=2.

I wrote out the integral $$\int\int 1+r^2 dr d\vartheta$$ integrated from 0 to $$2\pi$$ and r integrated from 0 to sqrt3 and volume enclosed gotten to be $$15\pi$$/2. So this is the volume enclosed by the hyperboloid and the xy-plane correct? So I wanted to use the volume of the cyclinder to subtract off the volume found previously. But the volume is $$6\pi$$ so if I subtract i would get negative volume?

It looks like you are integrating z^2 instead of z. Also shouldn't there be an extra r in the integral?

ie. $$\int\int z*r dr d\vartheta$$ whereas it looks like you are doing $$\int\int z^2 dr d\vartheta$$