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.
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.
is the area of the red triangle within the circle.
The blue area + the read area is one sixth the area of the circle.