Double integral bounded by a domain

Question

The integral I want to evaluate is $$\int \:\int _D\:\left(x+y\right)dxdy\:$$$$where\:D:\:xy\ge \:2,\:x+y\le \:3$$

I want to know how to find out the domain, I have no idea where to start from or how to make the graph even, any help is much appreciated!

To be clear, I just want to know whats the domain and how did you find it out in order to evaluate the double integral, the actual evaluation of it it's not needed since I know how to do it once I learnt the domain.

Answer 1

• $xy \ge 2 \implies (x > 0 \land y \ge 2/x) \lor (x < 0 \land y \le 2/x)$

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• $x+y \le 3 \implies y \le 3-x \implies$ the region below the line $y=3-x$.

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• Now, find the intersection of these two.

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So, as the value of $x$ ranges between $1$ and $2$, the value of $y$ is between $2/x$ and $3-x$. $$\iint\limits_D(x+y)dxdy = \int\limits_1^2 dx \int\limits_{2/x}^{3-x}(x+y)dy$$