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\problemgrouptitle{Areas Under a Curve}

\item Let $A$ be the area of the region $\mathcal{R}$ bounded by the $x$-axis, the lines $x=1$ and $x=4$, and the curve $f(x)=2x^2-x+2$ shown below:
\begin{center}
\resizebox{.4\textwidth}{!}{%
\includegraphics{graphics/area-under-a-curve1}%
}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item\label{itm:riemann} Write a formula for the $n^\text{th}$ sum $S_n$ approximating $A$ using right endpoints of the approximating rectangles.
\item Use $A=\displaystyle\lim_{n\to\infty}S_n$ and your answer from \ref*{itm:riemann} to calculate $A$.
\end{enumerate}
\answer{
\begin{enumerate}
\item $\displaystyle S_n=\frac{3}{n}\sum_{i=1}^nf\left(1+\displaystyle\frac{3i}{n}\right)=\sum_{i=1}^n\left(\frac{9}{n}+\frac{27i}{n^2}+\frac{54i^2}{n^3}\right)=\frac{81}{2}+\frac{81}{2n}+\frac{9}{n^2}$
\item $A=\displaystyle\lim_{n\to\infty}S_n=\frac{81}{2}$
\end{enumerate}
}

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