Changing the order of integration for $\int_{-t}^t \int_{z-a}^z \int_z^{y+a} f(x,y,z) d x d y d z$

Question

I'd like to change the order of integration in the following triple integral \begin{equation*} \int_{-t}^t \int_{z-a}^z \int_z^{y+a} f(x,y,z) d x d y d z \end{equation*} where $a > 0$ and $t > 0$. I'd like to integrate over $z$ first. The region of integration appears to be a prism; however, I'm having a hard time getting the correct limits.

Answer 1

Not a terribly good idea but you can do "blindly" via the inequalities $$z\le x\le y+a,$$ $$z-a\le y\le z,$$ $$-t\le z\le t.$$ The $z$ limits are obvious: $$y\le z\le x.$$ The upper limit for $x$ is also obviously $x\le y.$ Now, using $-t\le z$ and $z\le x$: $$-t\le x\le y.$$ Finally, $$-(t + a)\le y\le t.$$