September 6th, 2017, 11:27 AM  #1 
Senior Member Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics  Triple integral question
Hello folks. I have a question on this triple integral. $$\iiint_S z \,dx\,dy\,dz$$ where S is bounded by $x+y+z=2$, $x=0$, $y=0$, and $z=0$ I have the work on my white board. I got an answer of $\frac{5}{4}$ ... I sketched out a 3D coordinate system and noticed that this looks like a tetrahedron with Vertices (2,0,0), (0,2,0), and (0,0,2). However, I did not like how the integral was set up in the question, so I decided to use Fubini's theorem to write it in the form of $$\iiint_S z \,dz\,dy\,dx$$ so I could eventually write the limits like ... $$\int\limits_{0}^{1}\int\limits_{0}^{(2x)}\int\limits_{0}^{(2xy)} z \,dz\,dy\,dx$$ which lead me to the result of $\frac{5}{4}$. I just feel a little bit uncomfortable with switching around the differentials especially since the question had them set up in a particular way. Did I set this up correctly? Thank you! Jacob 
September 6th, 2017, 12:12 PM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,237 Thanks: 2412 Math Focus: Mainly analysis and algebra 
You can switch the "differentials" (they aren't really differentials) as long as you get the limits right. Why do you think $0 \le x \le 1$? I'd have guessed that it went up to 2. (Edit: now verified by a plot). Last edited by v8archie; September 6th, 2017 at 12:21 PM. 
September 6th, 2017, 02:05 PM  #3 
Senior Member Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics  Oh my god... What a silly, silly mistake. Thank you.

September 6th, 2017, 02:13 PM  #4  
Senior Member Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics  Quote:
Thanks  
September 6th, 2017, 02:17 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 18,721 Thanks: 1536  
September 6th, 2017, 03:51 PM  #6  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,237 Thanks: 2412 Math Focus: Mainly analysis and algebra  Quote:
 
September 6th, 2017, 03:53 PM  #7  
Senior Member Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics  Quote:
 
September 6th, 2017, 03:53 PM  #8 
Senior Member Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics  
September 6th, 2017, 06:05 PM  #9 
Senior Member Joined: Nov 2015 From: United States of America Posts: 165 Thanks: 21 Math Focus: Calculus and Physics  
September 7th, 2017, 02:34 AM  #10 
Global Moderator Joined: Dec 2006 Posts: 18,721 Thanks: 1536 
That looks okay. When you had 1 instead of 2, you shouldn't have got 5/4 (you missed a factor of 2 somewhere).


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