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I’m really having problems getting suitable limits when using double integration. For example:

Let $E$ be a region defined by

$E=\left\{(x,y): y-x \leq 2,\ x + y \geq 4,\ 2x + y \leq 8\right\}$

Sketch the region $E$.

The sketching is never a problem. So for this one we have:

$y = 8-2x$, $y=x+2$ and $y=4-x$ - the triangle between all the lines is the region $E$.

Find the area of $E$ and calculate

$\int\!\!\!\int_{E}\frac{1}{x}\,\mathrm{d} x\, \mathrm{d} y$

The integration isn’t the problem. I’m really confused as to how to get the correct limits. Our notes say to fix a variable and then look at the boundary of the other. So if we fix $x\in (1,4)$ then taking $y=x+2$ and $y=8-2x$ will give me a greater area then needed....

I hope someone can explain this to me as I’m rather confused!

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    can someone post a pic of the breakdown and explain how you get the boundaries/limits? its really confusing me!2011-05-30

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