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Simple volume calculation problem (double integrals) 
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#1
Jan313, 04:44 AM

P: 11

[EDIT]: Found the mistake, see the next post.
1. The problem statement, all variables and given/known data Evaluate $$\iint_{S}{\rm e}^{x+y}dx\, dy,S=\{(x,y):\leftx\right+\lefty\right\leq1\} $$ 2. The attempt at a solution ##\leftx\right+\lefty\right## is the rhombus with the center at the origin, symmetrical about both axes, so we find the volume at the first quadrant and then multiply by four. ##\varphi(x)=1x## limits the region of integration and so we have, first integrating over ##y## : $$\iint_{S}{\rm e}^{x+y}dx\, dy=4\int_{0}^{1}\left[\int_{0}^{1x}{\rm e}^{x}{\rm e}^{y}dy\right]dx=1\neq {\rm e}{\rm e}^{1}$$ (I know the correct answer which is ##{\rm e}{\rm e}^{1}## ) Apparently so simple, but I just can't see the mistake. 


#2
Jan313, 05:00 AM

P: 11

Oh Lord, ##{\rm e}^{x+y}## obviously isn't symmetric.



#3
Jan313, 07:34 PM

HW Helper
Thanks
PF Gold
P: 7,718




#4
Jan413, 01:55 AM

P: 11

Simple volume calculation problem (double integrals)
By the way, I'm learning this stuff on my own, so I might not know the standard way of approaching problems like this. My book (Apostol) says nothing about the change of variables if I understand correctly. What I dislike about this sort of examples is that I have to plot some of the functions first to actually understand what's going; TBH I'd enjoyed more analytic (or algorithmic) approach, where the peculiarities of a particular example wouldn't matter. (too many "I"'s in this post) 


#5
Jan413, 08:12 AM

Math
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Thanks
PF Gold
P: 39,682

No, that's not what LCKurtz meant by "the obvious change of variables".
If you let u= x+ y and v= x y, then x= (u+ v)/2 and y= (u v)/2. The figure is then [itex]1\le u\le 1[/itex], [itex]1\le v\le 1[/itex].The Jacobean is 1/2 so the integral becomes [tex]\frac{1}{2}\int_{u=1}^1\int_{v=1}^1 e^u dvdu[/tex] 


#6
Jan713, 03:01 AM

P: 11

Change of variables is some ten subsections below my current position in Apostol, but I liked the technique, thank you. 


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