Exercise: double integral of a surface

Let $f(x,y)=\sqrt{9-x^2-y^2}$ and $R=[0,1]\times[0,1]$. Compute the integral of $\sqrt{f_x^2+f_y^2}$ with Simpson $n=m=8$ and Gauss-Legendre $n=4$.

1. The partial derivatives are:

   $$\frac{\partial f}{\partial x}=-\frac{x}{\sqrt{9-x^2-y^2}},\qquad \frac{\partial f}{\partial y}=-\frac{y}{\sqrt{9-x^2-y^2}}$$

2. Therefore, the integrand simplifies to:

   $$\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}=\sqrt{\frac{x^2+y^2}{9-x^2-y^2}}$$

3. With double Simpson and n=m=8, one obtains:

   $$I_S=0.267814255559730$$

4. With Gauss-Legendre $n=4$, after transforming $x=\frac{u+1}{2}$ and $y=\frac{v+1}{2}$, one obtains:

   $$I_G=0.267770529696778$$