It is possible to show the expresssion
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: