Riemann sums

For each sum written in sigma notation, write the sum long-hand and evaluate the sum to find its value. For each sum written in expanded form, write the sum in sigma notation.

1. $\displaystyle \sum_{k=1}^{5} (k^2 + 2)$

2. $\displaystyle \sum_{i=3}^{6} (2i-1)$

3. $3 + 7 + 11 + 15 + \cdots + 27$

4. $4 + 8 + 16 + 32 + \cdots + 256$

5. $\displaystyle \sum_{i=1}^{6} \frac{1}{2^i}$

Suppose that an object moving along a straight line path has its velocity $v$ (in feet per second) at time $t$ (in seconds) given by .

1. Compute $M_5$, the middle Riemann sum, for $v$ on the time interval $[1,5]$. Be sure to clearly identify the value of $\Delta t$ as well as the locations of $t_0$, $t_1$, $\cdots$, $t_5$. In addition, provide a careful sketch of the function and the corresponding rectangles that are being used in the sum.

2. Buil…

Let $S$ be the sum given by .

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 sam…