Hi there, I have to compute the surface area for

$\displaystyle V:\{ -2(x+y)\leq{}z\leq{}4-x^2-y^2 \}$

I have a problem on finding the surface area for the paraboloid limited by the plane. I've parametrized the plane in polar coordinates, I thought it would be easier this way, but also tried in cartesian coordinates with the same result. The problem I have is to set the limits of integration on such a way that determines the area of the paraboloid limited by the plane.

So the paraboloid is parametrized by:

$\displaystyle \begin{Bmatrix}{ x=r\sin \theta}\\y=r \cos \theta \\z=4-r^2\end{matrix}

$

Then $\displaystyle T_r=( \sin \theata, \cos \theta,-2r);T_{\theta}=(r\cos\theta,-r\sin\theta,0)$

$\displaystyle T_r \times T_{\theta}=(-2r^2\sin^2 \theta,-2r^2\cos^2\theta,-r)$

$\displaystyle ||T_r \times T_{\theta}||=\sqrt[ ]{4r^4+r^2}$

Now the surface area is determined by: $\displaystyle \displaystyle\int_{D} ||T_r \times T_{\theta}||drd\theta$

Now D is the region inside the disc determined by the intersection of the surfaces. So D:

$\displaystyle -2x-2y=4-x^2-y^2\rightarrow (x-1)^2+(y-1)^2=6$

And this is my region of integration for my paraboloid. Now it doesn't seem so easy to express the region for the parametrization I choose.

So I'm trying to solve this in cartesian coordinates, this is the integral for the surface in cartesian coord:

$\displaystyle \displaystyle\int_{1-\sqrt[ ]{6}}^{1+\sqrt[ ]{6}}\int_{-\sqrt[ ]{6-(x-1)^2}+1}^{\sqrt[ ]{6-(x-1)^2}+1} \sqrt[ ]{4x^2+4y^2+1} dydx$

But this integral isn't so easy to solve. I tried to go from here to cylindrical coordinates, using the substitution $\displaystyle x=1+r\cos\theta,y=1+r\sin\theta$ that helps a bit with the limits of integration.

Then I get: $\displaystyle \displaystyle\int_{1-\sqrt[ ]{6}}^{1+\sqrt[ ]{6}}\int_{-\sqrt[ ]{6-(x-1)^2}+1}^{\sqrt[ ]{6-(x-1)^2}+1} \sqrt[ ]{4x^2+4y^2+1} dydx=\displaystyle\int_{0}^{\sqrt[ ]{6}}\int_{0}^{2\pi}r\sqrt[ ]{16+8r\cos\theta+8r\sin\theta+4r^2}d\theta dr$

Thats the best expression I get, but still too complicated to integrate by hand.