Integrate $G(x,y,z) = xyz$ over the triangular surface bounded by the point
$(1,0,0), (0,2,0)$ and $(0,1,1)$
Now I can calculate the equation of plane formed by the three points is :
$2x + y + z = 2$ but I am little confused on how should I calculate the limits for $x$ and $y$
Suppose I take the projection in $x-y$ plane then $z = 0$ then I get
$2x + y = 2$ So $x$ varies from $0$ to $1$ while $y$ varies from $0$ to $2-2x$
However in my book The limits of $y$ are given from $1-x$ to $2-2x$. I do not understand where this $1-x$ comes from ?
Can anyone please help me?
Thank you.
In the $xy$-plane, $y$ is bounded by the two lines. One of them is $y=2-2x$ and the other is the line passing through (1,0,0) and (0,1,0), which is the $xy$-projection of the side passing through (1,0,0) and (0,1,1).
Thus, the other limit is given by,
$$y = 1-x$$
i.e. the line passing through (1,0,0) and (0,1,0).