5.4 Integration formulas and the net change theorem (Page 2/8)

Calculus volume 1 Page 2 / 8

A graph with the x axis marked as t and the y axis marked normally. The lines y=40 and y=-30 are drawn over [2,4] and [4,5], respectively.The areas between the lines and the x axis are shaded. — The graph shows speed versus time for the given motion of a car.

Just as we did before, we can use definite integrals to calculate the net displacement as well as the total distance traveled. The net displacement is given by

\begin{matrix} \int_{2}^{5} v (t) d t & = \int_{2}^{4} 4 0 d t + \int_{4}^{5} −30 d t \\ = 80 - 30 \\ = 50 . \end{matrix}

Thus, at 5 p.m. the car is 50 mi north of its starting position. The total distance traveled is given by

\begin{matrix} \int_{2}^{5} | v (t) | d t & = \int_{2}^{4} 4 0 d t + \int_{4}^{5} 30 d t \\ = 80 + 30 \\ = 110 . \end{matrix}

Therefore, between 2 p.m. and 5 p.m., the car traveled a total of 110 mi.

To summarize, net displacement may include both positive and negative values. In other words, the velocity function accounts for both forward distance and backward distance. To find net displacement, integrate the velocity function over the interval. Total distance traveled, on the other hand, is always positive. To find the total distance traveled by an object, regardless of direction, we need to integrate the absolute value of the velocity function.

Finding net displacement

Given a velocity function $v (t) = 3 t - 5$ (in meters per second) for a particle in motion from time $t = 0$ to time $t = 3,$ find the net displacement of the particle.

Applying the net change theorem, we have

\begin{array}{l} \int_{0}^{3} (3 t - 5) d t & = \frac{3 t^{2}}{2} - 5 t |_{0}^{3} \\ = [\frac{3 {(3)}^{2}}{2} - 5 (3)] - 0 \\ = \frac{27}{2} - 15 \\ = \frac{27}{2} - \frac{30}{2} \\ = - \frac{3}{2} . \end{array}

The net displacement is $- \frac{3}{2}$ m ( [link] ).

A graph of the line v(t) = 3t – 5, which goes through points (0, -5) and (5/3, 0). The area over the line and under the x axis in the interval [0, 5/3] is shaded. The area under the line and above the x axis in the interval [5/3, 3] is shaded. — The graph shows velocity versus time for a particle moving with a linear velocity function.

Got questions? Get instant answers now!

Finding the total distance traveled

Use [link] to find the total distance traveled by a particle according to the velocity function $v (t) = 3 t - 5$ m/sec over a time interval $[0, 3] .$

The total distance traveled includes both the positive and the negative values. Therefore, we must integrate the absolute value of the velocity function to find the total distance traveled.

To continue with the example, use two integrals to find the total distance. First, find the t -intercept of the function, since that is where the division of the interval occurs. Set the equation equal to zero and solve for t . Thus,

\begin{matrix} 3 t - 5 & = & 0 \\ 3 t & = & 5 \\ t & = & \frac{5}{3} . \end{matrix}

The two subintervals are $[0, \frac{5}{3}]$ and $[\frac{5}{3}, 3] .$ To find the total distance traveled, integrate the absolute value of the function. Since the function is negative over the interval $[0, \frac{5}{3}],$ we have $| v (t) | = − v (t)$ over that interval. Over $[\frac{5}{3}, 3],$ the function is positive, so $| v (t) | = v (t) .$ Thus, we have

\begin{matrix} \int_{0}^{3} | v (t) | d t & = \int_{0}^{5 / 3} − v (t) d t + \int_{5 / 3}^{3} v (t) d t \\ = \int_{0}^{5 / 3} 5 - 3 t d t + \int_{5 / 3}^{3} 3 t - 5 d t \\ = {(5 t - \frac{3 t^{2}}{2}) |}_{0}^{5 / 3} + {(\frac{3 t^{2}}{2} - 5 t) |}_{5 / 3}^{3} \\ = [5 (\frac{5}{3}) - \frac{3 {(5 / 3)}^{2}}{2}] - 0 + [\frac{27}{2} - 15] - [\frac{3 {(5 / 3)}^{2}}{2} - \frac{25}{3}] \\ = \frac{25}{3} - \frac{25}{6} + \frac{27}{2} - 15 - \frac{25}{6} + \frac{25}{3} \\ = \frac{41}{6} . \end{matrix}

So, the total distance traveled is $\frac{14}{6}$ m.

Got questions? Get instant answers now!

Find the net displacement and total distance traveled in meters given the velocity function $f (t) = \frac{1}{2} e^{t} - 2$ over the interval $[0, 2] .$

Net displacement: $\frac{e^{2} - 9}{2} \approx - 0.8055 m;$ total distance traveled: $4 ln 4 - 7.5 + \frac{e^{2}}{2} \approx 1.740$ m

Got questions? Get instant answers now!

Applying the net change theorem

The net change theorem can be applied to the flow and consumption of fluids, as shown in [link] .

How many gallons of gasoline are consumed?

If the motor on a motorboat is started at $t = 0$ and the boat consumes gasoline at the rate of $5 - t^{3}$ gal/hr, how much gasoline is used in the first 2 hours?

Express the problem as a definite integral, integrate, and evaluate using the Fundamental Theorem of Calculus. The limits of integration are the endpoints of the interval $[0, 2] .$ We have

\begin{matrix} \int_{0}^{2} (5 - t^{3}) d t & = (5 t - \frac{t^{4}}{4}) |_{0}^{2} \\ = [5 (2) - \frac{{(2)}^{4}}{4}] - 0 \\ = 10 - \frac{16}{4} \\ = 6 . \end{matrix}

Thus, the motorboat uses 6 gal of gas in 2 hours.

Got questions? Get instant answers now!

Questions & Answers

how does Neisseria cause meningitis

Nyibol Reply

what is microbiologist

Muhammad Reply

what is errata

Muhammad

is the branch of biology that deals with the study of microorganisms.

Ntefuni Reply

What is microbiology

Mercy Reply

studies of microbes

Louisiaste

when we takee the specimen which lumbar,spin,

Ziyad Reply

How bacteria create energy to survive?

Muhamad Reply

Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments

_Adnan

But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?

Muhamad

they make spores

Louisiaste

what is sporadic nd endemic, epidemic

Aminu Reply

the significance of food webs for disease transmission

Abreham

food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.

Mark

explain assimilatory nitrate reduction

Esinniobiwa Reply

Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).

Elkana

This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products

Elkana

Examples of thermophilic organisms

Shu Reply

Give Examples of thermophilic organisms

Shu

advantages of normal Flora to the host

Micheal Reply

Prevent foreign microbes to the host

Abubakar

they provide healthier benefits to their hosts

ayesha

They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!

Mark

what is cell

faisal Reply

cell is the smallest unit of life

Fauziya

cell is the smallest unit of life

Akanni

Innocent

cell is the structural and functional unit of life

Hasan

is the fundamental units of Life

Musa

what are emergency diseases

Micheal Reply

There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️

_Adnan

define infection ,prevention and control

Innocent

I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life

Lubega

Heyy Lubega hussein where are u from?

_Adnan

en français

Adama

which site have a normal flora

ESTHER Reply

Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract

Safaa

skin

Asiina

skin,Oral,Nasal,GIt

Sadik

How can Commensal can Bacteria change into pathogen?

Sadik

How can Commensal Bacteria change into pathogen?

Sadik

all

Tesfaye

by fussion

Asiina

what are the advantages of normal Flora to the host

Micheal

what are the ways of control and prevention of nosocomial infection in the hospital

Micheal

what is inflammation

Shelly Reply

part of a tissue or an organ being wounded or bruised.

Wilfred

what term is used to name and classify microorganisms?

Micheal Reply

Binomial nomenclature

adeolu

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Calculus volume 1. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11964/1.2

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 1' conversation and receive update notifications?

Ask

	12 Biology 12 Mendel's Experiments Heredity MCQ By OpenStax Start Quiz
	1 Microeconomics 01 What Is Economics? By OpenStax Start Flashcards
	4 Physiotherapy Flashcards Set 4 By Rhodes Start Flashcards
©flickr: Luis	Atoms By Carly Allen Start Quiz
©flickr: Gareth	Resume Writing MCQ By Abby Sharp Start Quiz
	Anthropology Environment, Adaption Subsistence By Richley Crapo Start Assignment
	Anthropology Biology Culture By Richley Crapo Start Assignment
	12 Neuroanatomy 12 The Basal Ganglia By Stephen Voron Start Quiz
	Chemistry Final By Briana Hamilton Start Flashcards
©flickr: Donnie	How well do you know Beanie Boos? By Angela Eckman Start Quiz