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[EDIT]: Found the mistake, see the next post.

Evaluate $$\iint_{S}{\rm e}^{x+y}dx\, dy,S=\{(x,y):\left|x\right|+\left|y\right|\leq1\} $$

##\left|x\right|+\left|y\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)=1-x## limits the region of integration and so we have, first integrating over ##y## : $$\iint_{S}{\rm e}^{x+y}dx\, dy=4\int_{0}^{1}\left[\int_{0}^{1-x}{\rm e}^{x}{\rm e}^{y}dy\right]dx=1\neq {\rm e}-{\rm e}^{-1}$$ (I know the correct answer which is ##{\rm e}-{\rm e}^{-1}## )

Apparently so simple, but I just can't see the mistake.

## Homework Statement

Evaluate $$\iint_{S}{\rm e}^{x+y}dx\, dy,S=\{(x,y):\left|x\right|+\left|y\right|\leq1\} $$

**2. The attempt at a solution**##\left|x\right|+\left|y\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)=1-x## limits the region of integration and so we have, first integrating over ##y## : $$\iint_{S}{\rm e}^{x+y}dx\, dy=4\int_{0}^{1}\left[\int_{0}^{1-x}{\rm e}^{x}{\rm e}^{y}dy\right]dx=1\neq {\rm e}-{\rm e}^{-1}$$ (I know the correct answer which is ##{\rm e}-{\rm e}^{-1}## )

Apparently so simple, but I just can't see the mistake.

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