sketch the region of integration for $\int_0^2 \int_0^x \int_0^y f(x,y,z) \,dz\,dy\,dx$

Question

$$\int_0^2 \int_0^x \int_0^y f(x,y,z) \,dz\,dy\,dx$$

From what I can understand that the region should be between planes $x=0$, $x=2$ and $y=0$, $y=x$ and $z=0$, $z=y$.

I am finding it very difficult to represent it on a piece of paper. Please advise.

Answer 1

Yes you are right the limit to consider are

• $0\le x \le 2$ varing along $x$ axis
• $0\le y \le x$ varing in $x-y$ plane between $x$ axis and the line $y=x$
• $0\le z \le y$ varing in space between $x-y$ plane and the plane $z=y$

A good way to vizualize without a 3D plot program is try to make at first some sketch in the $x-y$, $y-z$, $x-z$ planes and then try to vizualize the region in 3D.

Here below an example

Answer 2

Here is the integration region:

Answer 3

Drawing cross-sections may be beneficial.

$x=0, x=2$ and $y=0, y=x$ define a right triangular prism, infinite in the $z$-direction. Draw its cross-section in the $z=c$ plane (I'd probably consider $c=0$ here for the $xy$-plane).

Now consider the intersections with $z=0$, if you haven't done so, and $z=y$. The former gives a right triangular base, and the latter cuts through your prism at a $\pi/4$ angle.

You could consider drawing the cross-section from another perspective as well - say choosing three projections flattening one of the cartesian components.