\input epsf
\nopagenumbers
\magnification=\magstep1

\noindent {\bf Problem statement} 
Define
$S_6(n) = 1^6 +2^6 + 3^6 + \cdots + n^6$
(in summation notation, $S_6(n) =
\sum\limits_{k=1}^n k^6$) then an explicit formula for $S_6(n)$ is
known, and it is:
$$S_6(n) = {1 \over {42}}\left(6n^7+21n^6+21n^5-7n^3+n\right)\,.$$

\noindent Similar formulas are known for other powers. These are
sometimes called, collectively, {\it Faulhaber's formula}. Jacob
Bernoulli also discovered these formulas but Faulhaber published
earlier. 
It isn't even clear that the values of this formula are
integers when $n$ is an integer! {\bf Assume that this formula is
true.}

\medskip

\noindent a) Check the formula for $S_6$ by evaluating it for $n=4$.
The answer should be the same as $1^6+2^6+3^6+4^6$.

\medskip

\noindent b) Find some area and some approximating sum for this area
which knowledge of this formula will allow you to evaluate exactly.
Write the approximating sums, and evaluate the limit of these sums as
$n\to\infty$ to compute the area.










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