Hello, KK!
List five other iterated integrals that equals to:
. .
I write in the limits "completely".
. .
I start with the "outside" limits: .
That is, ranges from to Code:
y

  1+     
::::::::::::::
::::::::::::::
::::::::::::::
+ x

The next limits are: .
That is, ranges between the line and Code:
y

  1+    * 
 /:
 /:::
 /:::::
/+ x
 1
Now lay this diagram "on the floor"; the zaxis goes straight up. Code:
z



    * y
*::*
*:::::*
*::::::::*
* * * * * * *
/
x
I can't draw this next part . . . hope you can follow.
With no restriction on , we have a triangular prism
. . with the above triangle as a crosssection, extending up and down infinitely.
But the limits are: and
. . That is, ranges between the "floor" and the slanted plane
So our triangular prism is cut off at floorlevel below
. . and the slanted plane above.
And that is the solid we are dealing with.
Now if we change the order of integration, the limits are changed.
With the first of the answers, we have:
We will try to describe the same solid using this order of limits.
Starting outside: goes from to
Then: goes from to
Then: goes from to
So the integral is: .
. . I hope this helps . . .