I'm having trouble describing a region in space.
Let $B$ the region bounded by the planes $x=0$, $y=0$, $z=0$, $x+y=1$, $z=x+y$. One way of calculating the volume of B is:
\begin{equation}
V(B)=\int_{0}^{1} \int_{0}^{1-x} \int_{0}^{x+y}dzdydx=\frac{1}{3}
\end{equation}
I'm trying to change the order to integration to $\int \int \int dxdydz$, but I can't get my integration limits to work (I get different values for the volume).
I thought that $(0<z<1)$, and $(z-y<x<1-y)$, but I can't find the limits on the Y axis. I believe it sould be something like:
\begin{equation}
\int_0^1\int_?^?\int_{z-y}^{1-y}dxdydz
\end{equation}
Best Answer
Try to work it out from the hints before reading the spoilers.
The first hint is to make a sketch of the cross-section of a plane $z=c$ with the volume of interest and $c$ some constant for which $0 \leq c \leq 1$. What is the shape of this cross-section?