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!