5.1 Approximating areas  (Page 2/17)

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Write in sigma notation and evaluate the sum of terms 2 i for $i=3,4,5,6.$

$\sum _{i=3}^{6}{2}^{i}={2}^{3}+{2}^{4}+{2}^{5}+{2}^{6}=120$

The properties associated with the summation process are given in the following rule.

Rule: properties of sigma notation

Let ${a}_{1},{a}_{2}\text{,…,}\phantom{\rule{0.2em}{0ex}}{a}_{n}$ and ${b}_{1},{b}_{2}\text{,…,}\phantom{\rule{0.2em}{0ex}}{b}_{n}$ represent two sequences of terms and let c be a constant. The following properties hold for all positive integers n and for integers m , with $1\le m\le n.$

1. $\sum _{i=1}^{n}c=nc$

2. $\sum _{i=1}^{n}c{a}_{i}=c\sum _{i=1}^{n}{a}_{i}$

3. $\sum _{i=1}^{n}\left({a}_{i}+{b}_{i}\right)=\sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}$

4. $\sum _{i=1}^{n}\left({a}_{i}-{b}_{i}\right)=\sum _{i=1}^{n}{a}_{i}-\sum _{i=1}^{n}{b}_{i}$

5. $\sum _{i=1}^{n}{a}_{i}=\sum _{i=1}^{m}{a}_{i}+\sum _{i=m+1}^{n}{a}_{i}$

Proof

We prove properties 2. and 3. here, and leave proof of the other properties to the Exercises.

2. We have

$\begin{array}{cc}\sum _{i=1}^{n}c{a}_{i}\hfill & =c{a}_{1}+c{a}_{2}+c{a}_{3}+\text{⋯}+c{a}_{n}\hfill \\ & =c\left({a}_{1}+{a}_{2}+{a}_{3}+\text{⋯}+{a}_{n}\right)\hfill \\ \\ \\ & =c\sum _{i=1}^{n}{a}_{i}.\hfill \end{array}$

3. We have

$\begin{array}{cc}\sum _{i=1}^{n}\left({a}_{i}+{b}_{i}\right)\hfill & =\left({a}_{1}+{b}_{1}\right)+\left({a}_{2}+{b}_{2}\right)+\left({a}_{3}+{b}_{3}\right)+\text{⋯}+\left({a}_{n}+{b}_{n}\right)\hfill \\ & =\left({a}_{1}+{a}_{2}+{a}_{3}+\text{⋯}+{a}_{n}\right)+\left({b}_{1}+{b}_{2}+{b}_{3}+\text{⋯}+{b}_{n}\right)\hfill \\ \\ \\ & =\sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}.\hfill \end{array}$

A few more formulas for frequently found functions simplify the summation process further. These are shown in the next rule, for sums and powers of integers , and we use them in the next set of examples.

Rule: sums and powers of integers

1. The sum of n integers is given by
$\sum _{i=1}^{n}i=1+2+\text{⋯}+n=\frac{n\left(n+1\right)}{2}.$
2. The sum of consecutive integers squared is given by
$\sum _{i=1}^{n}{i}^{2}={1}^{2}+{2}^{2}+\text{⋯}+{n}^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}.$
3. The sum of consecutive integers cubed is given by
$\sum _{i=1}^{n}{i}^{3}={1}^{3}+{2}^{3}+\text{⋯}+{n}^{3}=\frac{{n}^{2}{\left(n+1\right)}^{2}}{4}.$

Evaluation using sigma notation

Write using sigma notation and evaluate:

1. The sum of the terms ${\left(i-3\right)}^{2}$ for $i=1,2\text{,…,}\phantom{\rule{0.2em}{0ex}}200.$
2. The sum of the terms $\left({i}^{3}-{i}^{2}\right)$ for $i=1,2,3,4,5,6.$
1. Multiplying out ${\left(i-3\right)}^{2},$ we can break the expression into three terms.
$\begin{array}{cc}\sum _{i=1}^{200}{\left(i-3\right)}^{2}\hfill & =\sum _{i=1}^{200}\left({i}^{2}-6i+9\right)\hfill \\ \\ \\ & =\sum _{i=1}^{200}{i}^{2}-\sum _{i=1}^{200}6i+\sum _{i=1}^{200}9\hfill \\ & =\sum _{i=1}^{200}{i}^{2}-6\sum _{i=1}^{200}i+\sum _{i=1}^{200}9\hfill \\ & =\frac{200\left(200+1\right)\left(400+1\right)}{6}-6\left[\frac{200\left(200+1\right)}{2}\right]+9\left(200\right)\hfill \\ & =2,686,700-120,600+1800\hfill \\ & =2,567,900\hfill \end{array}$
2. Use sigma notation property iv. and the rules for the sum of squared terms and the sum of cubed terms.
$\begin{array}{cc}\sum _{i=1}^{6}\left({i}^{3}-{i}^{2}\right)\hfill & =\sum _{i=1}^{6}{i}^{3}-\sum _{i=1}^{6}{i}^{2}\hfill \\ \\ \\ \\ & =\frac{{6}^{2}{\left(6+1\right)}^{2}}{4}-\frac{6\left(6+1\right)\left(2\left(6\right)+1\right)}{6}\hfill \\ & =\frac{1764}{4}-\frac{546}{6}\hfill \\ & =350\hfill \end{array}$

Find the sum of the values of $4+3i$ for $i=1,2\text{,…,}\phantom{\rule{0.2em}{0ex}}100.$

15,550

Finding the sum of the function values

Find the sum of the values of $f\left(x\right)={x}^{3}$ over the integers $1,2,3\text{,…,}\phantom{\rule{0.2em}{0ex}}10.$

Using the formula, we have

$\begin{array}{cc}\sum _{i=0}^{10}{i}^{3}\hfill & =\frac{{\left(10\right)}^{2}{\left(10+1\right)}^{2}}{4}\hfill \\ \\ & =\frac{100\left(121\right)}{4}\hfill \\ & =3025.\hfill \end{array}$

Evaluate the sum indicated by the notation $\sum _{k=1}^{20}\left(2k+1\right).$

440

Approximating area

Now that we have the necessary notation, we return to the problem at hand: approximating the area under a curve. Let $f\left(x\right)$ be a continuous, nonnegative function defined on the closed interval $\left[a,b\right].$ We want to approximate the area A bounded by $f\left(x\right)$ above, the x -axis below, the line $x=a$ on the left, and the line $x=b$ on the right ( [link] ).

How do we approximate the area under this curve? The approach is a geometric one. By dividing a region into many small shapes that have known area formulas, we can sum these areas and obtain a reasonable estimate of the true area. We begin by dividing the interval $\left[a,b\right]$ into n subintervals of equal width, $\frac{b-a}{n}.$ We do this by selecting equally spaced points ${x}_{0},{x}_{1},{x}_{2}\text{,…,}\phantom{\rule{0.2em}{0ex}}{x}_{n}$ with ${x}_{0}=a,{x}_{n}=b,$ and

${x}_{i}-{x}_{i-1}=\frac{b-a}{n}$

for $i=1,2,3\text{,…,}\phantom{\rule{0.2em}{0ex}}n.$

We denote the width of each subinterval with the notation Δ x , so $\text{Δ}x=\frac{b-a}{n}$ and

${x}_{i}={x}_{0}+i\text{Δ}x$

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