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Question

# Let W be the region bounded by the planes $x = 0 , y = 0 , z = 0 , x + y = 1$ , and $z = x + y$ (a) Find the volume of W. (b) Evalute $\displaystyle \iiint _ { W } x d x d y d z$ (c) Evalute $\displaystyle \iiint _ { W } y d x d y d z$

Solution

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$\textbf{Projection}$ of $W$ on $xy$-plane is a triangle with sides $x=0$, $y=0$ and $x+y=1$ (see the $\text{\color{Sepia}picture}$ below).This triangle can be represented as

$\begin{equation*} 0\leq x\leq 1,\quad 0\leq y\leq 1-x. \end{equation*}$

Therefore, we may conclude that the region $W$ can be described by

$$$\boxed{0\leq x\leq 1,\quad 0\leq y\leq 1-x,\quad 0\leq z\leq x+y}$$$

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