Deriving the formula of the volume of a sphere using integration I'm trying to derive the formula for the volume of a sphere, using integration :
$\int_{0}^{\pi r}\pi r^{2}dc$
$\pi r^{2}$ is constant, so :
$\pi r^{2}\int_{0}^{\pi r}dc$
Integrating, I get only c, and plug $\pi r$ for c.
So I'm getting $\pi^{2}r^{3}$, instead of $\frac{4}{3}\pi r^{3}$.
My reasoning is that I start with a full disc ($\pi r^{2}$), then rotating the disc for half the circumference of the sphere ($\pi r$) to get the final volume.
What's wrong with my reasoning ?

Your integral gives the volume of a cylinder of he hight $\pi*r$ your dc would be better called dh . You do not rotate the circle!
