Additional Discussion of Newton's formula

It is possible to show the expression in a rather informal way with the help of the diagram above.

Claim : The definite integral represents the blue area in the above cirle.
Let , then

which is the locus of the above circle with radius unit and center at .
It follows that the definite integral from x=0 to of the function is equal to the blue area in the circle shown above. Claim: is the area of the red triangle within the circle.

Observation : The blue area + the read area is one sixth the area of the circle.