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.

Deduction: