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Why is the volume of a sphere $\frac{4}{3}\pi r^3$?
We know that the surface area of a sphere is $4\pi r^2$ and the volume is $\frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere.
How does one prove these formulae?
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... and see also http://math.stackexchange.com/questions/625/why-is-the-derivative-of-a-circles-area-its-perimeter-and-similarly-for-sphere?rq=1 – Hans Lundmark Oct 21 '12 at 15:16
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For the area, use the equation of a circle of radius $r$, $x^2+y^2=r^2$, to find the area between two curves.
For the volume, view the sphere of radius $r$ as a solid of revolution of the function $y=\sqrt{r^2-x^2}$.
BobaFret
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In this answer I give a geometric derivation for the surface area of a sphere, then integrate that by shells to get the volume of the sphere.
