TITLE: Volume of $E=\{5x^8 \le y \le 7x^8; 2y^5 \le z \le 3y^5; z^7 \le x \le 6z^7\}$.
QUESTION [6 upvotes]: I am stuck with the following problem:
Compute volume of $E=\{5x^8 \le y \le 7x^8; 2y^5  \le z  \le 3y^5; z^7 \le x \le 6z^7\}$.
My progress:
It is easy to see that $x,y,z \ge 0$. Therefore I can add all inequalities and after simple algebra get the following: $$5x^8 + 2y^5+z^7 \le x+y+z \le 7x^8 + 3y^5+6x^7$$
Update: as it was pointed out by A.Γ. adding inequalities does not work because different sets can give the same inequalities. So, it looks like I should solve or at least get some kind of information for integration directly from the system somehow. 
Any hint?
Thanks a lot for your help!

REPLY [4 votes]: Hint: the set is given by 
$$
\begin{cases}
5\le\frac{y}{x^8}\le 7,\\
2\le\frac{z}{y^5}\le 3,\\
1\le\frac{x}{z^7}\le 6
\end{cases}\quad\rightarrow\quad\text{substitution}\quad\rightarrow\quad
\begin{cases}
5\le u\le 7,\\
2\le v\le 3,\\
1\le w\le 6
\end{cases}.
$$