I believe you're still looking for something to find your mistake with. Now your things aren't looking that bad from here so I guess just giving you the solution will let you see where you're wrong in your numbers crunching.

Let $S = \{ T(r,\theta) = (r \cos \theta, r \sin \theta, r^2 (\cos^2 \theta - \sin^2 \theta)) \, | \, \theta \in [0,2\pi], r \in [0,1] \}$ be the chosen parametrization of $S$. Therefore

$\phi(T(r,\theta)) = (4(r^2(\cos^2 \theta - \sin^2 \theta)) + 8 (r \sin \theta)^2 + 1)^{3/2} $

$ = (4r^2 \cos^2 \theta - 4 r^2 \sin^2 \theta + 8 r^2 \sin^2 \theta + 1)^{3/2} = (4r^2 + 1)^{3/2}$

and

$
T_r \times T_{\theta} = 
\begin{vmatrix}
\vec i &amp; \vec j &amp; \vec k \\
\cos \theta &amp; \sin \theta &amp; 2r(\cos^2 \theta - \sin^2 \theta) \\
- r \sin \theta &amp; r \cos \theta  &amp; r^2(-4 \sin \theta \cos \theta)
\end{vmatrix}
$

$
= (-4r^2\sin^2 \theta \cos \theta - 2r^2 \cos^3 \theta  + 2r^2 \sin^2 \theta \cos \theta, 4r^2 \sin \theta \cos^2 \theta - 2r^2 \sin \theta \cos^2 \theta + 2r^2 \sin^3 \theta, r)
$

$
=(-2r^2 \cos \theta, 2 r^2 \sin \theta,r) \quad
$
so that $\quad ||T_r \times T_{\theta}|| = \sqrt{4r^4 + r^2} = r \sqrt{4r^2 + 1}$.
Now it's easy because
$$
\int_0^{2\pi} \int_0^1 (4r^2+1)^{3/2} (r\sqrt{4r^2 +1}) \, dr d\theta = 2\pi \int_0^1 r(4r^2 + 1)^2 \, dr
$$
which gives a number I am not in the mood for computing right now, but I think you'll be fine. =)

Problem calculating an integral over a surface
Have 12 excel Sheets in a workbook for each month Jan-Dec.  Each sheet contains Names and leaves marked for that month in the same row and sum value for each category of leave at the end of the table.  I need to summarize these individual sheets value by looking for each person's Name in all sheets. how can I achieve this in excel....  I tried sum of vlookup from all sheet, it throws error or no result.  
