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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: