# Rewrite the triple integral

Rewrite the triple integral $$\int_0^1\int_0^x\int_0^y f(x,y,z)\,dz\,dy\,dx$$ as $$\int_a^b\int_{g_1(z)}^{g_2(z)}\int_{h_1(yz)}^{h_2(y,z)}\,dx\,dy\,dz$$ I found $a=0$ , $b=1$, $g_2(z)=1$, $h_2(y,z)=1$, so I need to find $g_1(z)$ and $h_1(y,z)$.

• In addition to my answer below, you might find my answer to a related problem about finding the limits of integration for double integrals after changing the order of integration helpful. – David H Mar 14 '14 at 2:44

$$0 \leq z \leq y,\\ 0 \leq y \leq x,\\ 0 \leq x \leq 1.$$
$$0 \leq z \leq y \leq x \leq 1.$$
This representation of the domain of integration has the advantage of describing the domain of integration in a more natural geometric manner that doesn't presuppose any order of integration. After choosing a different order of integration, it is ideal for deriving the appropriate bounds for a new equivalent triple of inequalities analogous to the triple above. Since the new order of integration is $dxdydz$, we find:
$$y \leq x \leq 1,\\ z \leq y \leq 1,\\ 0 \leq z \leq 1.$$