I want to compute the integral $$\int_{M} yz\,\,\, dS$$ where $M=\{(x,y,z):x,y,z\geq 0, x+y+z-1=0\}$.
I tried it as follows:
Let us consider the chart $$\phi:[0,1]^2\rightarrow M,\,\, \phi(x,y)=(x,y,1-x-y)$$ Now I computed $$D\phi=\left(\begin{array}{rrr}1&0\\0&1\\-1&-1\end{array}\right)$$ Then the gram matris is $$G_\phi=\left(\begin{array}{rrr} 2&1\\1&2\end{array}\right)$$ which impies that the gram determinant $g_\phi=3$.
Then I computed \begin{align} \int_0^1\int_0^1 y(1-x-y)\sqrt 3 dxdy& =\sqrt{3}\int_0^1 (xy-\frac{1}{2}x^2y-xy^2|_{x=0}^1)dy\\ &=\sqrt{3}\int_0^1\frac{1}{2}y-y^2 dy\\ &=-\frac{\sqrt{3}}{12}. \end{align}
But I think this is wrong since we have a negative value or did I do it correct?
Could someone take a look at it?