evaluation of surface integral over the unit disk and plane in $xy$ axis

Evaluation of $$\int_{S}\int(x^2+y^2+z^2)dS$$

Where $S:z=x+y,x^2+y^2\leq 1$

What i try: Let $g(x,y,z)=x^2+y^2+z^2$ and $-1\leq x,y\leq 1$

Let $S(x,y)=<x,y,x+y>$

Then $S_{x}(x,y)=<1,0,1>$ and $S_{y}=<0,1,1>$

So $||S_{x}\times S_{y}||=|-\vec{i}-\vec{j}+\vec{k}|=\sqrt{3}$

So $$\int_{S}\int g(x,y,z)dS=\int\int g(x,y,z)|S_{x}\times S_{y}|dA$$

$$=\sqrt{3}\int^{1}_{-1}\int^{1}_{-1}\bigg(x^2+y^2+(x+y)^2\bigg)dxdy$$

Please explain me is my integral is right. If not how do i form it. Help me please

1 Answer

The bounds indicate that $x^2 + y^2 \leq 1$ hence this is the unit disk in $\mathbb{R}^2$. You're best to convert to polar coordinates in the last step:

$$ \iint_S \sqrt{3}\left (x^2 + y^2 + (x+y)^2\right ) dxdy \;\; =\;\; \sqrt{3}\int_0^{2\pi}\int_0^1 r\left (2r^2 + r^2\sin(2\theta)\right )drd\theta. $$