I've gotten this result a couple of times trying to "derive" basic geometric area/volume equations and I thought I had resolved it internally already. However, I was thinking back again to finding the volume of a sphere and can't understand why this is incorrect for the volume of a sphere: $(\pi r)(\pi r^{2}) = \pi^{2}r^{3}$ Here is an image to 'justify' my intuition: Circle Rotation
The image shows a 180 degree rotation. We know the circumference of a circle is $2\pi r$ which we can divide in half for half a rotation. We can then multiply that by the area $\pi r^{2}$ to find the volume. However, of course this is wrong, but why? We know the actual volume to be $\frac{4}{3}\pi r^{3}$. I know that Archimedes was able to derive this geometrically, so I shouldn't need calculus to find this.