Graphing a triple integral, software and the actual integral

Question

What software can be used to graph a triple integral? I tried to use wolfram alpha, but it wouldn't work.

I am trying to graph:$ \int_0^1 \int_\sqrt{z}^1 \int_0^{2-y-z} f(x,y,z) \; \mathrm{dx\; dy\; dz}$

Could anyone graph this for me please, and tell me the used software! Thank you for your help!!!

Answer 1

Without a specified integrand $f(x,y,z)$, I presume you just want to plot the region of integration corresponding to the integral; i.e., $$R = \{(x,y,z) \in \mathbb R^3 \mid (0 \le z \le 1) \wedge (\sqrt{z} \le y \le 1) \wedge (0 \le x \le 2-y-z) \}.$$ We can equivalently write this as $$R = \{(x,y,z) \in \mathbb R^3 \mid (0 \le z \le 1) \wedge (z \le y^2 \le 1) \wedge (0 \le x) \wedge (x+y+z \le 2)\}.$$ This region consists of the following boundaries:

1. The plane $z = 0$.
2. The plane $x = 0$.
3. The plane $y = 1$.
4. The plane $x+y+z = 2$.
5. The parabolic cylinder $z = y^2$.

The plane $z = 1$ is redundant, because it is automatically satisfied with the second condition $\sqrt{z} \le y \le 1$. To sketch this region, plot each of the five boundaries.