# 5.1 Approximating areas  (Page 7/17)

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1. Find an upper sum for $f\left(x\right)=10-{x}^{2}$ on $\left[1,2\right];$ let $n=4.$
2. Sketch the approximation.
1. $\text{Upper sum}=8.0313.$

2. ## Finding lower and upper sums for $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$

Find a lower sum for $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ over the interval $\left[a,b\right]=\left[0,\frac{\pi }{2}\right];$ let $n=6.$

Let’s first look at the graph in [link] to get a better idea of the area of interest. The graph of y = sin x is divided into six regions: Δ x = π / 2 6 = π 12 .

The intervals are $\left[0,\frac{\pi }{12}\right],\left[\frac{\pi }{12},\frac{\pi }{6}\right],\left[\frac{\pi }{6},\frac{\pi }{4}\right],\left[\frac{\pi }{4},\frac{\pi }{3}\right],\left[\frac{\pi }{3},\frac{5\pi }{12}\right],$ and $\left[\frac{5\pi }{12},\frac{\pi }{2}\right].$ Note that $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ is increasing on the interval $\left[0,\frac{\pi }{2}\right],$ so a left-endpoint approximation gives us the lower sum. A left-endpoint approximation is the Riemann sum $\sum _{i=0}^{5}\text{sin}\phantom{\rule{0.1em}{0ex}}{x}_{i}\left(\frac{\pi }{12}\right).$ We have

$\begin{array}{cc}A\hfill & \approx \text{sin}\left(0\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{12}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{6}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{4}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{\pi }{3}\right)\left(\frac{\pi }{12}\right)+\text{sin}\left(\frac{5\pi }{12}\right)\left(\frac{\pi }{12}\right)\hfill \\ & =0.863.\hfill \end{array}$

Using the function $f\left(x\right)=\text{sin}\phantom{\rule{0.1em}{0ex}}x$ over the interval $\left[0,\frac{\pi }{2}\right],$ find an upper sum; let $n=6.$

$A\approx 1.125$

## Key concepts

• The use of sigma (summation) notation of the form $\sum _{i=1}^{n}{a}_{i}$ is useful for expressing long sums of values in compact form.
• For a continuous function defined over an interval $\left[a,b\right],$ the process of dividing the interval into n equal parts, extending a rectangle to the graph of the function, calculating the areas of the series of rectangles, and then summing the areas yields an approximation of the area of that region.
• The width of each rectangle is $\text{Δ}x=\frac{b-a}{n}.$
• Riemann sums are expressions of the form $\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)\text{Δ}x,$ and can be used to estimate the area under the curve $y=f\left(x\right).$ Left- and right-endpoint approximations are special kinds of Riemann sums where the values of $\left\{{x}_{i}^{*}\right\}$ are chosen to be the left or right endpoints of the subintervals, respectively.
• Riemann sums allow for much flexibility in choosing the set of points $\left\{{x}_{i}^{*}\right\}$ at which the function is evaluated, often with an eye to obtaining a lower sum or an upper sum.

## Key equations

• Properties of Sigma Notation
$\sum _{i=1}^{n}c=nc$
$\sum _{i=1}^{n}c{a}_{i}=c\sum _{i=1}^{n}{a}_{i}$
$\sum _{i=1}^{n}\left({a}_{i}+{b}_{i}\right)=\sum _{i=1}^{n}{a}_{i}+\sum _{i=1}^{n}{b}_{i}$
$\sum _{i=1}^{n}\left({a}_{i}-{b}_{i}\right)=\sum _{i=1}^{n}{a}_{i}-\sum _{i=1}^{n}{b}_{i}$
$\sum _{i=1}^{n}{a}_{i}=\sum _{i=1}^{m}{a}_{i}+\sum _{i=m+1}^{n}{a}_{i}$
• Sums and Powers of Integers
$\sum _{i=1}^{n}i=1+2+\text{⋯}+n=\frac{n\left(n+1\right)}{2}$
$\sum _{i=1}^{n}{i}^{2}={1}^{2}+{2}^{2}+\text{⋯}+{n}^{2}=\frac{n\left(n+1\right)\left(2n+1\right)}{6}$
$\sum _{i=0}^{n}{i}^{3}={1}^{3}+{2}^{3}+\text{⋯}+{n}^{3}=\frac{{n}^{2}{\left(n+1\right)}^{2}}{4}$
• Left-Endpoint Approximation
$A\approx {L}_{n}=f\left({x}_{0}\right)\text{Δ}x+f\left({x}_{1}\right)\text{Δ}x+\text{⋯}+f\left({x}_{n-1}\right)\text{Δ}x=\sum _{i=1}^{n}f\left({x}_{i-1}\right)\text{Δ}x$
• Right-Endpoint Approximation
$A\approx {R}_{n}=f\left({x}_{1}\right)\text{Δ}x+f\left({x}_{2}\right)\text{Δ}x+\text{⋯}+f\left({x}_{n}\right)\text{Δ}x=\sum _{i=1}^{n}f\left({x}_{i}\right)\text{Δ}x$

State whether the given sums are equal or unequal.

1. $\sum _{i=1}^{10}i$ and $\sum _{k=1}^{10}k$
2. $\sum _{i=1}^{10}i$ and $\sum _{i=6}^{15}\left(i-5\right)$
3. $\sum _{i=1}^{10}i\left(i-1\right)$ and $\sum _{j=0}^{9}\left(j+1\right)j$
4. $\sum _{i=1}^{10}i\left(i-1\right)$ and $\sum _{k=1}^{10}\left({k}^{2}-k\right)$

a. They are equal; both represent the sum of the first 10 whole numbers. b. They are equal; both represent the sum of the first 10 whole numbers. c. They are equal by substituting $j=i-1.$ d. They are equal; the first sum factors the terms of the second.

In the following exercises, use the rules for sums of powers of integers to compute the sums.

$\sum _{i=5}^{10}i$

$\sum _{i=5}^{10}{i}^{2}$

$385-30=355$

Suppose that $\sum _{i=1}^{100}{a}_{i}=15$ and $\sum _{i=1}^{100}{b}_{i}=-12.$ In the following exercises, compute the sums.

$\sum _{i=1}^{100}\left({a}_{i}+{b}_{i}\right)$

$\sum _{i=1}^{100}\left({a}_{i}-{b}_{i}\right)$

$15-\left(-12\right)=27$

$\sum _{i=1}^{100}\left(3{a}_{i}-4{b}_{i}\right)$

$\sum _{i=1}^{100}\left(5{a}_{i}+4{b}_{i}\right)$

$5\left(15\right)+4\left(-12\right)=27$

In the following exercises, use summation properties and formulas to rewrite and evaluate the sums.

$\sum _{k=1}^{20}100\left({k}^{2}-5k+1\right)$

$\sum _{j=1}^{50}\left({j}^{2}-2j\right)$

$\sum _{j=1}^{50}{j}^{2}-2\sum _{j=1}^{50}j=\frac{\left(50\right)\left(51\right)\left(101\right)}{6}-\frac{2\left(50\right)\left(51\right)}{2}=40,\text{​}375$

$\sum _{j=11}^{20}\left({j}^{2}-10j\right)$

$\sum _{k=1}^{25}\left[{\left(2k\right)}^{2}-100k\right]$

$4\sum _{k=1}^{25}{k}^{2}-100\sum _{k=1}^{25}k=\frac{4\left(25\right)\left(26\right)\left(51\right)}{9}-50\left(25\right)\left(26\right)=-10,\text{​}400$

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