$$ \int xyz \ \ dV$$ over domain T Where T is a tetrahedron with vertices $$ (0,0,0),(1,0,0) ,(1,1,0) \ and\\ (1,0,1) $$ is a type I region capped by the planes = and the plane through the points (,,),(,,) and (,,) (whose equation can be found to be ( – + + = 0). The domain D is the triangle with vertices (,,),(,,),(,,)
My thoughts that it can be done with three types triple integral is my thought correct and if not could you please explain to me why not?
I don't know what you mean by “type I region” or “three types triple integral”, but if you make the linear change of variables $$ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = u \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + v \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + w \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} , $$ then you get a tetrahedron in $uvw$-space with corners at $(0,0,0)$, $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$, which should be easy to handle.