Integration Involving the Absolute Function

Question

How do I integrate the double integral of the form $|x^2-y|$ with the boundaries $-1\leq x\leq 1$ and $-1\leq y\leq 1$?

Answer 1

You can break it into two integrals, one inside the parabola where $y \gt x^2$ and one outside where $y \lt x^2$ So $$\int_{-1}^1\int_{-1}^1 |x^2-y|\; dy \; dx=\\ \int_{-1}^1\int_{-1}^{x^2}x^2-y \;dy \; dx+\int_{-1}^1\int_{x^2}^{1}y-x^2 \;dy \; dx$$