Why does the volume of a spheres formula conist of 4/3?
There are two ways to calculate the volume of the sphere. The first way is by using calculus in a Cartesian coordinate system. We can imagine the sphere as a collection of disks of different sizes glued together. By finding the total volume of all the disks, we will know the volume of the sphere. Let us first set out our coordinates. We define the origin to be at the center of the sphere. We assume that each disk has a thickness of dx, which is a very very small distance (we call it infinitesimal), and a radius of y. We can find the value of y based on the distance of the disk from the side of the sphere. The volume of one such thin disk is π y² dx. To find the volume of the sphere, we add up all the volumes of the thin disks. We can do this by integration. Volume = ∫ π y² dx, from x=-R to x=R But since r is a constant, x and y are related by Pythagorus’ theorem. r²=x²+y² We can now replace the y² term in the integration. Volume = ∫ π (r²-x²) dx, from x=-R to x=R After integration: V=π
