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\begin{document}

\begin{questions}

\question % workshop bank Ch. 5 -> esergel51
Let $g(x) = \begin{cases} x &\hbox{if } x>1 \\ c &\hbox{if } x=1 \\ 1 &\hbox{if } x<1 \end{cases}$.

\begin{parts}

\part Compute the third right-endpoint approximation $R_3$ for the area under $g(x)$ between $a=0$ and $b=3$. Your answer will depend on $c$. What is $R_3$ when $c=7$? When $c=8$? How much did your answer change when you changed $c$?

\part Compute $R_4$. Your answer will not depend on $c$.

\part Compute $R_6$ for the same area. Your answer will depend on $c$. What is $R_6$ when $c=7$? When $c=8$? How much did your answer change?

\part Set up, but don't compute, and expression for $R_{300}$ of the same area. What is the difference between $R_{300}$ when $c=7$ and $R_{300}$ when $c=8$?

\part Suppose $c=1$. Now $g(x)$ is continuous on $[0,3]$. What is $\int_{0}^{3} g(x) \, dx$?

\part Even if $c \neq 1$, we still say $\int_{0}^{3} g(x) \, dx$ equals the value you computed in part (e). That is, the value of $g(x)$ at a single point doesn't matter when computing a definite integral. Explain why this makes sense with Riemann approximations. [Hint: How many terms of $R_N$ does $c$ affect? What happens to these terms as $N \to \infty$?]

\end{parts}

\end{questions}

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