§4.2 Riemann sums

Introduction

Sigma Notation

Riemann Sums

When the function is sometimes negative

Expand and evaluate the sigma sum $\sum_{k=1}^{5}(k^2 + 2)$ term by term.

Expand and evaluate the sigma sum $\sum_{i=3}^{6}(2i - 1)$ term by term.

The sum $3 + 7 + 11 + 15 + \cdots + 27$ jumps by four each step. In sigma notation it is

The sum $4 + 8 + 16 + 32 + \cdots + 256$ doubles each step. In sigma notation it is

For a positive increasing function on $[a, b]$, the left sum $L_n$ and right sum $R_n$ satisfy

For $v(t) = 0.25t^3 - 1.5t^2 + 3t + 0.25$ on $[0,2]$ with four rectangles, the left diagram samples $v(0) = 0.25$, $v(0.5) = 1.40625$, $v(1) = 2$, $v(1.5) = 2.21875$ with width $0.5$. The sum equals

To capture the distance an object travels on $2 \le t \le 5$, the region to sketch is

For the same walker, $L_4 = \frac{311}{48}$, $R_4 = \frac{335}{48}$, and $M_4 = \frac{637}{96}$. Averaging $L_4$ and $R_4$

Computing midpoint sums for a velocity on $[1,5]$ gives $M_{10} = -1.36$ and $M_{20} = -1.34$. As $n$ grows without bound the sums approach

For the walker's velocity table, the right endpoint sum with four rectangles of width $0.5$ is $R_4 = \frac{335}{48}$. Compute its decimal value, rounded to three decimals.

A velocity function sampled at midpoints on $[1,5]$ with five rectangles gives $M_5 = -\frac{36}{25}$. Report this estimate of the total change in position, rounded to two decimals.

The same midpoint samples carry absolute values $\frac{7}{25}, \frac{17}{25}, 1, \frac{17}{25}, \frac{7}{25}$, each times width $\frac{4}{5}$. Compute the estimate of total distance traveled, rounded to three decimals.

The sum $\sum_{i=1}^{6}\frac{1}{2^i}$ equals $\frac{63}{64}$, which is strictly less than $1$.