# Setting up the triple integral in $\mathrm {dz dy dx}$ order

Question:- Find the volume of the region bounded by $$x+z=8 ,z=x ,y=8, z=y$$ and $$z=0$$.

The region looks like a prism with vertical base( $$y=8$$) on one side and slanted base( $$z=y$$) on other side.

$$V=\int_0^4 \int_z^8 \int_z^{8-z}\mathrm {dx dy dz}=\frac{320}{3}$$

But when I evaluate in the order $$\mathrm {dz dy dx}$$, I can see that we have to write volume as sum of some triple integrals, I was able to write integrals only for $$4\le y \le 8$$.In the region $$0\le y \le 4$$, It is difficult to get z bounds as the line perpendicular to $$x-y$$ plane intersects given planes in some region, whereas in other region it doesn't.

$$V=\int_0^4 \int_4^8 \int_0^x\mathrm {dz dy dx}+\int_4^8 \int_4^8 \int_0^{8-x}\mathrm {dz dy dx}+...$$

If you are integrating in the order $$dz~ dy ~ dx$$, focus on the projection of the region in xy-plane.

The projection of the region bound by $$z = x, z + x = 8, z = y$$ and $$y = 8$$ in xy-plane is a square bound by $$x = 0, x = 8, y = 0$$ and $$y = 8$$. Also, the projection of the intersection of $$z = y$$ with planes $$z = x$$ and $$z + x = 8$$ are lines $$y = x$$ and $$x + y = 8$$ in xy-plane. So for part of the region bound by lines $$x = y, x + y = 8, y = 0$$ in xy-plane, $$z$$ is bound above by plane $$z = y$$. For the part of the projection outside of the triangular region and inside the square, $$z$$ is bound above either by $$z = x$$ for $$0 \lt x \lt 4$$ and by $$z +x = 8$$ for $$4 \lt x \lt 8$$.

So to summarize,
a) For $$0 \lt x \lt 4$$,
$$i)$$ $$0 \lt y \lt x$$, $$0 \lt z \lt y$$
$$ii)$$ $$x \lt y \lt 8$$, $$0 \lt z \lt x$$

b) For $$4 \lt x \lt 8$$,
$$i)$$ $$0 \lt y \lt 8 - x, 0 \lt z \lt y$$
$$ii)$$ $$8 - x \lt y \lt 8, 0 \lt z \lt 8 - x$$

Using symmetry you can find volume of $$(a)$$ and multiply by $$2$$ or evaluate both.

$$\displaystyle V = 2 \int_0^4 \int_0^x \int_0^y dz ~dy~ dx + 2 \int_0^4 \int_x^8 \int_0^x dz~ dy ~dx = \frac{320}{3}$$

• Thankyou very much, you just cleared my 2 days confusion. Dec 31, 2021 at 13:12
• @user1055 glad I could help Dec 31, 2021 at 13:23