Question: Suppose $f$ is integrable. Express $\int^1_0 \int^{2y}_y \int^{x+y}_0 f(x,y,z)\ dz\ dx\ dy$ as a sum of iterated integrals in the order $\ dx \ dy \ dz$.

My Attempt: Since $z=x+y$, $z_{max}=x_{max}+y_{max}=2y_{max}+1=2(1)+1=3$ and $z_{min}=x_{min}+y_{min}=y_{min}+0=0$. Therefore, $0\le z \le 3$ and $0 \le x+y \le 3$.

I've used the last inequality to set up a graph of the bounds in the $x$-$y$ plane, and can see how I can use this to write up a sum of iterated integrals, however they'll only be in terms of $x$ and $y$ which does not satisfy the question.

Can anyone help me from here?



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