Fig. 1

By recalling the general formula for power summation derived in “Little Bird and a Recursive Generator“, namely

We can obtain the result

where and , without the Fundamental Theorem of Calculus.

To this end, we divide the interval into sub-intervals of equal length as shown in Fig. 1. Let denotes the sum of the areas of the rectangles. The value tends to as increases, is the definite integral of from to , i.e.,

.

Since

It follows from (1) that

.

For reduces to

,

and the result immediately follows:

.

For , we let

.

Clearly,

.

Since ,

.

i.e.,

.

Therefore,

.

Applying this result to the area from to* * we have

,

and by subtraction of the areas,

.