The integral of the circumference of a circle is its area: $$\int 2\pi*r = \pi *r ^2$$
Likewise, does the integral of area have physical significance? $$\int \pi *r ^2 = \frac{\pi*r^3 }{3}$$
(Its similar to the volume of a sphere, but its only 1/4 of that, so that seems coincidence. )
It is the volume of the cone with height equal to $r$. You have originally your disk made out of increasing radius rings. In this case you have a cone made out of increasing radius disks