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\noindent Problems esergel52, esergel53, and esergel54 are intended to be used together.

\begin{questions}


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\question % workshop bank Ch. 5 -> esergel54
(Bonus problem, not to be turned in) Recall the formula
$$
\sum_{j=1}^N j^2 = \frac{N(N+1)(2N+1)}{6}
$$
Our goal is to compute the $N$-th left- and right-endpoint approximations $L_N$ and $R_N$ for the area under the curve $f(x)=x^2$ between $x=0$ and $x=b$ when $b>0$.

\begin{parts}

\part What is $\Delta x$? What is $x_j$? These should depend on $b$ and $N$.

\part Set up a formula for $R_N$ as a sum. Then rearrange the terms so that you can use the formula above to get an expression for $R_N$ which has no $\sum$ or $\dots$ in it.

\part Write a formula for $\displaystyle \sum_{j=0}^{N-1} j^2$. Check that your formula works when $N=4$.

\part Set up a formula for $L_N$ as a sum. Then use the formula from part (c) to get an expression for $L_N$ which has no $\sum$ or $\dots$ in it.

\part Now take the limit of the expressions you got in parts (b) and (d) as $N \to \infty$. Check that you get $\frac{b^3}{3}$ for both of these limits.

\end{parts}

\end{questions}

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