Exercise 9

Let $S$ be the sum given by $$\begin{gathered} S = ((1.4)^2 + 1) \cdot 0.4 + ((1.8)^2 + 1) \cdot 0.4 + ((2.2)^2 + 1) \cdot 0.4 + ((2.6)^2 + 1) \cdot 0.4 +((3.0)^2 + 1) \cdot 0.4 \end{gathered}$$.

1. Assume that $S$ is a right Riemann sum. For what function $f$ and what interval $[a,b]$ is $S$ this function's Riemann sum? Why?

2. How does your answer to (a) change if $S$ is a left Riemann sum? a middle Riemann sum?

3. Suppose that $S$ really is a right Riemann sum. What is geometric quantity does $S$ approximate?

4. Use sigma notation to write a new sum $R$ that is the right Riemann sum for the same function, but that uses twice as many subintervals as $S$.