When solving some exercises, I forgot the formula for the surface area of the unit sphere. However, I remember that the length of the perimeter of a circle of radius $r$ is $2 \pi r$. So I figured I'll just compute
$$2 \int_0^1 2 \pi r \, \mathrm dr,$$
starting to integrate from the north pole of a unit sphere up to its center and as that area occurs twice, I added a factor of $2$. However, this integral yields $2 \pi$ and not the desired $4 \pi$. Where did I forget another factor of $2$?
I tried doing the same computation with the volume of the sphere since its surface area will just be the derivative of the volume. However, again
$$2 \int_0^1 \pi r^2 \, \mathrm dr = \frac{2}{3} \pi,$$
missing a factor of $2$. What am I doing wrong?