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**HallsofIvy** $\displaystyle \int_{y= 0}^3\int_{x= 0}^{3y- y^2} \sqrt{x^2+ y^2} dxdy$

$\displaystyle x= 3y- y^2$ is a parabola with horizontal axis. It crosses the y-axis at (0, 0) and (0, 3). Completing the square, $\displaystyle x= \frac{9}{4}- (y- \frac{3}{2})^2$ so that $\displaystyle y- \frac{3}{2}= \pm\sqrt{x- \frac{9}{4}}$. That tells us that x ranges from 0 to $\displaystyle \frac{9}{4}$ and that, for each x, y ranges from $\displaystyle \frac{3}{2}- \sqrt{x- \frac{9}{4}}$ to $\displaystyle \frac{3}{2}+ \sqrt{x- \frac{9}{4}}$.