Problem: Find the center of gravity of a regular cone whose height is h and the radius of its circular base is r, assuming that it is made of a homogeneous material.

Solution: The key to a successful solution here is a proper placement of the cone:

Now we see that the cone is equal to the solid obtained by revolving (about the x-axis) the graph of the line y = rx/h for x between 0 and h. In this setting, the center of gravity lies on the x-axis, therefore its position is determined by finding its x-coordinate. We use the appropriate formula and the fact that the solid is homogeneous, that is, the density is constant and it cancels in the formula:

Since the placement of the cone was just a tool for calculation, we should try to express the answer in geometrical terms. For instance, we can say that the center of mass lies on the axis of symmetry of the cone at the height h/4 above the base.

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