If you are willing to accept that the volume of a cone is ⅓ Bh where B is the area of the base and if you are willing to accept that the area of the surface of the sphere is 4 pi r^2 then you can get the volume of the sphere by adding an infinite number of thin cones radiating from the center with height = r. The sum of the B values is the area of the surface of the sphere giving ⅓ r * 4 pi r^2 = 4/3 pi r^3.

You know that a circle is π r ². (I can prove that if you don’t believe me.)

So if you integrate the area of a circle π r ² from -r to r, where r is the circular curve sqrt(r ² – x ²), you get 4/3 π r^3.

@Vortico

You’re variable of integration should be something like r-prime, but otherwise that’s correct.

@Vortico Oh, oops! I meant to say the variable of integration is x.

@Vortico

No worries, I meant to say your.