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Geometric forumlas related as integrals

I noticed today that the formulas for the area and circumference of a circle are related as integrals. The same goes for the the volume and surface of a cube. In other words:

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Also, when I draw the graphs of the two formulas for the circle, they intersect when r=2. With the sphere they intersect at r=3. Does this stem from the fact that the two objects have 2 and 3 dimensions, respectively?

I think I would understand integrals a whole lot better if I knew how this all fits together. So can anyone explain, or at least point me in the right direction?

Thanks(Hi)

Edit: mods, can you correct my spelling in the title? I obviously meant formulas.