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I'm just a little unsure of how to tackle this one. I understand that typically you would separate the integral into two for where x is positive or negative, I'm just unsure of how to separate it for both x and y.

The integral is $\int_{-1}^{1}\int_{-2(x+1)^2}^{2(x-1)^2} |x+y| dydx$

Thanks in advance :)

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You are first integrating with respect to $y$ so you could split it into one part where $x+y$ is non-negative, implying $y\geq -x$ and an other part where $x+y$ is negative implying $y<-x$ and thus for this second part $|x+y|=-(x+y)$.

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