# 1.2 The definite integral  (Page 6/16)

 Page 6 / 16
$\frac{89+90+56+78+100+69}{6}=\frac{482}{6}\approx 80.33.$

Therefore, your average test grade is approximately 80.33, which translates to a B− at most schools.

Suppose, however, that we have a function $v\left(t\right)$ that gives us the speed of an object at any time t , and we want to find the object’s average speed. The function $v\left(t\right)$ takes on an infinite number of values, so we can’t use the process just described. Fortunately, we can use a definite integral to find the average value of a function such as this.

Let $f\left(x\right)$ be continuous over the interval $\left[a,b\right]$ and let $\left[a,b\right]$ be divided into n subintervals of width $\text{Δ}x=\left(b-a\right)\text{/}n.$ Choose a representative ${x}_{i}^{*}$ in each subinterval and calculate $f\left({x}_{i}^{*}\right)$ for $i=1,2\text{,…,}\phantom{\rule{0.2em}{0ex}}n.$ In other words, consider each $f\left({x}_{i}^{*}\right)$ as a sampling of the function over each subinterval. The average value of the function may then be approximated as

$\frac{f\left({x}_{1}^{*}\right)+f\left({x}_{2}^{*}\right)+\text{⋯}+f\left({x}_{n}^{*}\right)}{n},$

which is basically the same expression used to calculate the average of discrete values.

But we know $\text{Δ}x=\frac{b-a}{n},$ so $n=\frac{b-a}{\text{Δ}x},$ and we get

$\frac{f\left({x}_{1}^{*}\right)+f\left({x}_{2}^{*}\right)+\text{⋯}+f\left({x}_{n}^{*}\right)}{n}=\frac{f\left({x}_{1}^{*}\right)+f\left({x}_{2}^{*}\right)+\text{⋯}+f\left({x}_{n}^{*}\right)}{\frac{\left(b-a\right)}{\text{Δ}x}}.$

Following through with the algebra, the numerator is a sum that is represented as $\sum _{i=1}^{n}f\left({x}_{i}^{*}\right),$ and we are dividing by a fraction. To divide by a fraction, invert the denominator and multiply. Thus, an approximate value for the average value of the function is given by

$\begin{array}{cc}\frac{\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)}{\frac{\left(b-a\right)}{\text{Δ}x}}\hfill & =\left(\frac{\text{Δ}x}{b-a}\right)\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)\hfill \\ \\ & =\left(\frac{1}{b-a}\right)\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)\text{Δ}x.\hfill \end{array}$

This is a Riemann sum. Then, to get the exact average value, take the limit as n goes to infinity. Thus, the average value of a function is given by

$\frac{1}{b-a}\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}f\left({x}_{i}\right)\text{Δ}x=\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)dx.$

## Definition

Let $f\left(x\right)$ be continuous over the interval $\left[a,b\right].$ Then, the average value of the function $f\left(x\right)$ (or f ave ) on $\left[a,b\right]$ is given by

${f}_{\text{ave}}=\frac{1}{b-a}{\int }_{a}^{b}f\left(x\right)dx.$

## Finding the average value of a linear function

Find the average value of $f\left(x\right)=x+1$ over the interval $\left[0,5\right].$

First, graph the function on the stated interval, as shown in [link] .

The region is a trapezoid lying on its side, so we can use the area formula for a trapezoid $A=\frac{1}{2}h\left(a+b\right),$ where h represents height, and a and b represent the two parallel sides. Then,

$\begin{array}{cc}{\int }_{0}^{5}x+1dx\hfill & =\frac{1}{2}h\left(a+b\right)\hfill \\ & =\frac{1}{2}·5·\left(1+6\right)\hfill \\ & =\frac{35}{2}.\hfill \end{array}$

Thus the average value of the function is

$\frac{1}{5-0}{\int }_{0}^{5}x+1dx=\frac{1}{5}·\frac{35}{2}=\frac{7}{2}.$

Find the average value of $f\left(x\right)=6-2x$ over the interval $\left[0,3\right].$

3

## Key concepts

• The definite integral can be used to calculate net signed area, which is the area above the x -axis less the area below the x -axis. Net signed area can be positive, negative, or zero.
• The component parts of the definite integral are the integrand, the variable of integration, and the limits of integration.
• Continuous functions on a closed interval are integrable. Functions that are not continuous may still be integrable, depending on the nature of the discontinuities.
• The properties of definite integrals can be used to evaluate integrals.
• The area under the curve of many functions can be calculated using geometric formulas.
• The average value of a function can be calculated using definite integrals.

## Key equations

• Definite Integral
${\int }_{a}^{b}f\left(x\right)dx=\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}f\left({x}_{i}^{*}\right)\text{Δ}x$
• Properties of the Definite Integral
${\int }_{a}^{a}f\left(x\right)dx=0$
${\int }_{b}^{a}f\left(x\right)dx=\text{−}{\int }_{a}^{b}f\left(x\right)dx$
${\int }_{a}^{b}\left[f\left(x\right)+g\left(x\right)\right]dx={\int }_{a}^{b}f\left(x\right)dx+{\int }_{a}^{b}g\left(x\right)dx$
${\int }_{a}^{b}\left[f\left(x\right)-g\left(x\right)\right]dx={\int }_{a}^{b}f\left(x\right)dx-{\int }_{a}^{b}g\left(x\right)dx$
${\int }_{a}^{b}cf\left(x\right)dx=c{\int }_{a}^{b}f\left(x\right)$ for constant c
${\int }_{a}^{b}f\left(x\right)dx={\int }_{a}^{c}f\left(x\right)dx+{\int }_{c}^{b}f\left(x\right)dx$

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!