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Volume of a sphere
We want to calculate the volume
of a sphere
with radius
.
Approximating the sphere with thin cylinders of heights
, for
, and approaching the limit, we get
. Substituting
and
, gives
^{3/2})
.
Volume of a sphere
We want to calculate the volume [tex']V[/tex'] of a sphere [tex']S^2 \subset \mathbb{R}^3[/tex'] with radius [tex']r[/tex'].
Approximating the sphere with thin cylinders of heights [tex']2\sqrt{r^2-t^2}[/tex'], for [tex']t \in [0,r][/tex'], and approaching the limit, we get [tex']V = \int_0^r 2\pi t 2\sqrt{r^2-t^2}dt[/tex']. Substituting [tex']u = r^2 - t^2[/tex'] and [tex']\frac{du}{dt} = -2t dt[/tex'], gives [tex']V = -2\pi\int_{r^2}^0 \sqrt{u}du[/tex'] [tex']=2\pi\frac{2}{3}(r^2)^{3/2}[/tex'] [tex']=\frac{4}{3}\pi r^3[/tex'].
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