Root of the problem

Find the mass and center of mass of the solid E with the given density function .

This is one of the interesting problems because the conventional first step is to visualize the problem and define the boundaries which will be used later for integration. My instinct is to eliminate any radical sign and express the condition in a more standard form. By squaring equations with radical signs, we can normally see these equations transform into a clearer visual representation of parabola, circle or sphere on some axes.

Although that routine works well most of the time, I found out that it can be misleading also sometimes. The bounding curve for instance in the problem becomes a regular parabola when squared but when left alone, which is best, it shows precisely what it wants: only the curve along the positive y-axis or the positive root of x.

Without heeding this clue, it is a headache to point out the target region in this otherwise simple problem. So, the problem basically hinges on the root of x.

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