INTEGRATING ROUNDING FUNCTIONS-(III)

24 03 2009

SQUARE FLOOR FUNCTION DEFINITE INTEGRAL:

\displaystyle I_3= \int_0^x \lfloor x \rfloor^2 dx = \int_0^{\lfloor x \rfloor} \lfloor x \rfloor^2 dx+ \int_{{\lfloor x \rfloor}}^ {x} \lfloor x \rfloor^2 dx

\displaystyle I_3=\sum_{k=1}^{ \lfloor x \rfloor-1}{k^2} + \left\{{x}\right\}\lfloor x \rfloor^2

\displaystyle I_3=P(\lfloor x \rfloor-1) + \left\{{x}\right\}\lfloor x \rfloor^2

Where: \displaystyle P(n) gives the n-th Square Pyramidal Number.

\displaystyle P(n) =\frac{n(n+1)(2n+1)}{6}

POWER FLOOR FUNCTION DEFINITE INTEGRAL:

\displaystyle I_4= \int_0^x \lfloor x \rfloor^n dx = \sum_{k=1}^{ \lfloor x \rfloor-1}{k^n} + \left\{{x}\right\}\lfloor x \rfloor^n

\displaystyle S(n,m)=\sum_{k=1}^{m}{k^n} \;\;\;\; is the Faulhaber’s formula.

If \displaystyle n=1 , the formula gives the Triangular Numbers.

And if \displaystyle n=2 , the formula gives the Square Pyramidal Numbers.

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