This question is a peculiar one, to say the least.

For $\displaystyle b>0$, let $\displaystyle P$ be the partition of $\displaystyle [0,b]$ into $\displaystyle n$ equal subintervals. Calculate both $\displaystyle S_P$ and $\displaystyle s_P$ for $\displaystyle f(x)=x^3$ and prove that

$\displaystyle J(=glb(S_P))\leq 1/4b^4(1+1/n)^2$ and that $\displaystyle I(=lub(s_P))\geq 1/4b^4(1-1/n)^2$

for any positive $\displaystyle n$.

Deduce that $\displaystyle f(x)$ is integrable on $\displaystyle [0,b]$ and $\displaystyle \int_0^b x^3dx=1/4b^4$.

NOTE: $\displaystyle \sum_{k=1}^n k^3=1/4n^2(n+1)^2$

EDIT: I found some help on the question, though his answer is a little long. I'll post a link below to his work.

http://answers.yahoo.com/question/in...8081532AALd5im

If he made any mistakes, be sure to point them out.