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Let's consider a function: $f(x,y) = k(x^2+y^2)$, where $0\le\lvert x\rvert\le y\le 1$

To find the value of $k$, I need to set up a double integral with the correct domain, i.e $0\le\lvert x\rvert\le y\le 1$ (since this is a probability question).

My main problem is setting up the domain, since the region given contains $\lvert x\rvert$. My attempt is as follows:

$0 \le \lvert x\rvert\le y == 0\le x \le y, -y\le x \le 0$
So the double integral will be: $\int_{0}^1 \int_{-y}^0 f(x,y) dxdy + \int_{0}^1 \int_{0}^y f(x,y) dxdy$.

Am I doing this right? I would appreciate if someone could assess my working. Thanks!

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    This is correct! Indeed, you can simplify this a little: $0\le|x|$ is automatic, so $|x|\le y$ is equivalent to $-y\le x\le y$, and your two double integrals can be combined into $\int_{0}^1 \int_{-y}^y f(x,y) \,dx\,dy$. – 2017-01-20
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    That makes more sense. Thanks! – 2017-01-20

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