Get all integration orders and plot them?

Questions
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1

Hello,
I just downloaded Wolfram Mathematica and would like to know how to plot and get the integration limits in all orders. I know it can be done because I saw someone doing it in college and he was using Mathematica, probably an older version though. The volume described by the integrals are not defined using functions but as the sort:

integral.jpg

I went through the references and guides and could find regionplot3D function and others similar but all of them need the range for x,y and z. I’d like to be able to put the inequalities such as in the image and get all 6 possibles integrals to calculate. I don’t know if i’m being clear :s

Also can I use wolfram cloud on my phone using the college’s license? I tried to but apparently I have to pay for mathematica online (and app therefore). Can this (what I’m asking) be done in wolfram alpha or Matlab?

Thanks and excuse my english

2

Perhaps

Integrate[1, {x, 0, 1}, {z, 0, x}, {y, x/2, x}]
3

Thanks for the answer but I don’t wish to calculate the integral , just write it in different orders. I found out the Reduce function to be the closest to what I’m looking for:

Screenshot_6.jpg

The red rectangle represents the right answer for the dydxdz order (it integrates from the outside to the inside apparently) but how can I do to clean the excess useless conditions?
Moreover, the teacher that did this, only typed the inequalities and it automatically plotted and gave him the integrals, amazing.
Thanks

4

This doesn’t answer your question, but here is one way to just type in the inequalities and have Mathematica plot and integrate:

region = ImplicitRegion[x/2 <= y <= x && 0 <= z <= x && x <= 1, {x, y, z}];

DiscretizeRegion[region]
Integrate[1, {x, y, z} \[Element] region]