# Converting an iterated integral into a **sum** of iterated integrals

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?