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Math Help - Wierd reationship in spheres

  1. #1
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    Wierd reationship in spheres

    I found a strange relationship between a sphere's volume and area, and a circle's area and circumference.

    For spheres:
    \frac{d}{dr}V=\frac{d}{dr}\frac{4}{3}\pi r^3=4\pi r^2=A
    Where V is the volume and A is the area.

    For cirles:
    \frac{d}{dr}A=\frac{d}{dr}\pi r^2=2\pi r=O
    Where A is the area and O is the circumference.

    I'm sure that this applies to higher dimensions as well.

    What gives rise to this relation ship? Is it a "coincidence"?

    Thanks in advance.
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  2. #2
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    The volume of the sphere is defined by summing infinitely many thin slices of the sphere. The radius of each disc or circle is y=\sqrt{r^2-x^2} and the surface area of each circle is therefore \pi y^2.

    For a sphere centered on the origin, we want the circles to be summed from -r to r so

    V=\int_{-r}^r \pi (r^2-x^2)dx = \frac{4}{3}\pi r^3
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  3. #3
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    Yes, that's true. But what does it have to do with the question?
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