Double integrals in polar coordinates

Hello, can someone please provide any help to the following problem for an exam on Saturday.

Use polar coordinates to find the volume of the given solid:

a) Below the parabloid z = 18 – 2x^2 – 2y^2 and above the xy-plane

b) Enclosed by the hyperboloid – x^2 – y^2 + z^2 = 1 and the plane z = 2

Any help will be appreciated. Thank you.

Re: Double integrals in polar coordinates

I suggest using cylindrical polar coordinates for the first. It is bounded by the plane $\displaystyle \displaystyle z = 0$ and the paraboloid $\displaystyle \displaystyle z = 18 - 2x^2 - 2y^2 = 2(9 - x^2 - y^2) = 2(9 - r^2\cos^2{\theta} - r^2\sin^2{\theta}) = 2(9 - r^2\cos{2\theta}) $

So $\displaystyle \displaystyle 0 \leq z \leq 2(9 - r^2\cos{2\theta})$.

At any $\displaystyle \displaystyle z$ co-ordinate, a circle is traced out by $\displaystyle \displaystyle x$ and $\displaystyle \displaystyle y$. It should be clear that the radius $\displaystyle \displaystyle 0 \leq r \leq 3$ (can you see why?) and it is swept out across all angles, so $\displaystyle \displaystyle 0 \leq \theta \leq 2\pi$ (can you see why?)

Can you set up the triple integral now?