<< Chapter < Page Chapter >> Page >
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

2 5 v ( t ) d t = 2 4 4 0 d t + 4 5 −30 d t = 80 30 = 50 .

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

2 5 | v ( t ) | d t = 2 4 4 0 d t + 4 5 30 d t = 80 + 30 = 110 .

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

0 3 ( 3 t 5 ) d t = 3 t 2 2 5 t | 0 3 = [ 3 ( 3 ) 2 2 5 ( 3 ) ] 0 = 27 2 15 = 27 2 30 2 = 3 2 .

The net displacement is 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!
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,

3 t 5 = 0 3 t = 5 t = 5 3 .

The two subintervals are [ 0 , 5 3 ] and [ 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 , 5 3 ] , we have | v ( t ) | = v ( t ) over that interval. Over [ 5 3 , 3 ] , the function is positive, so | v ( t ) | = v ( t ) . Thus, we have

0 3 | v ( t ) | d t = 0 5 / 3 v ( t ) d t + 5 / 3 3 v ( t ) d t = 0 5 / 3 5 3 t d t + 5 / 3 3 3 t 5 d t = ( 5 t 3 t 2 2 ) | 0 5 / 3 + ( 3 t 2 2 5 t ) | 5 / 3 3 = [ 5 ( 5 3 ) 3 ( 5 / 3 ) 2 2 ] 0 + [ 27 2 15 ] [ 3 ( 5 / 3 ) 2 2 25 3 ] = 25 3 25 6 + 27 2 15 25 6 + 25 3 = 41 6 .

So, the total distance traveled is 14 6 m.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

Net displacement: e 2 9 2 0.8055 m; total distance traveled: 4 ln 4 7.5 + e 2 2 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

0 2 ( 5 t 3 ) d t = ( 5 t t 4 4 ) | 0 2 = [ 5 ( 2 ) ( 2 ) 4 4 ] 0 = 10 16 4 = 6 .

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

what is the power rule
Vanessa Reply
how do i deal with infinity in limits?
Itumeleng Reply
Add the functions f(x)=7x-x g(x)=5-x
Julius Reply
f(x)=7x-x g(x)=5-x
Awon
5x-5
Verna
what is domain
Cabdalla Reply
difference btwn domain co- domain and range
Cabdalla
x
Verna
The set of inputs of a function. x goes in the function, y comes out.
Verna
where u from verna
Arfan
If you differentiate then answer is not x
Raymond
domain is the set of values of independent variable and the range is the corresponding set of values of dependent variable
Champro
what is functions
mahin Reply
give different types of functions.
Paul
how would u find slope of tangent line to its inverse function, if the equation is x^5+3x^3-4x-8 at the point(-8,1)
riyad Reply
pls solve it i Want to see the answer
Sodiq
ok
Friendz
differentiate each term
Friendz
why do we need to study functions?
abigail Reply
to understand how to model one variable as a direct relationship to another variable
Andrew
integrate the root of 1+x²
Rodgers Reply
use the substitution t=1+x. dt=dx √(1+x)dx = √tdt = t^1/2 dt integral is then = t^(1/2 + 1) / (1/2 + 1) + C = (2/3) t^(3/2) + C substitute back t=1+x = (2/3) (1+x)^(3/2) + C
navin
find the nth differential coefficient of cosx.cos2x.cos3x
Sudhanayaki Reply
determine the inverse(one-to-one function) of f(x)=x(cube)+4 and draw the graph if the function and its inverse
Crystal Reply
f(x) = x^3 + 4, to find inverse switch x and you and isolate y: x = y^3 + 4 x -4 = y^3 (x-4)^1/3 = y = f^-1(x)
Andrew
in the example exercise how does it go from -4 +- squareroot(8)/-4 to -4 +- 2squareroot(2)/-4 what is the process of pulling out the factor like that?
Robert Reply
can you please post the question again here so I can see what your talking about
Andrew
√(8) =√(4x2) =√4 x √2 2 √2 hope this helps. from the surds theory a^c x b^c = (ab)^c
Barnabas
564356
Myong
can you determine whether f(x)=x(cube) +4 is a one to one function
Crystal
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
one to one means that every input has a single output, and not multiple outputs. whenever the highest power of a given polynomial is odd then that function is said to be odd. a big help to help you understand this concept would be to graph the function and see visually what's going on.
Andrew
can you show the steps from going from 3/(x-2)= y to x= 3/y +2 I'm confused as to how y ends up as the divisor
Robert Reply
step 1: take reciprocal of both sides (x-2)/3 = 1/y step 2: multiply both sides by 3 x-2 = 3/y step 3: add 2 to both sides x = 3/y + 2 ps nice farcry 3 background!
Andrew
first you cross multiply and get y(x-2)=3 then apply distribution and the left side of the equation such as yx-2y=3 then you add 2y in both sides of the equation and get yx=3+2y and last divide both sides of the equation by y and you get x=3/y+2
Ioana
Multiply both sides by (x-2) to get 3=y(x-2) Then you can divide both sides by y (it's just a multiplied term now) to get 3/y = (x-2). Since the parentheses aren't doing anything for the right side, you can drop them, and add the 2 to both sides to get 3/y + 2 = x
Melin
thank you ladies and gentlemen I appreciate the help!
Robert
keep practicing and asking questions, practice makes perfect! and be aware that are often different paths to the same answer, so the more you familiarize yourself with these multiple different approaches, the less confused you'll be.
Andrew
please how do I learn integration
aliyu Reply
they are simply "anti-derivatives". so you should first learn how to take derivatives of any given function before going into taking integrals of any given function.
Andrew
best way to learn is always to look into a few basic examples of different kinds of functions, and then if you have any further questions, be sure to state specifically which step in the solution you are not understanding.
Andrew
example 1) say f'(x) = x, f(x) = ? well there is a rule called the 'power rule' which states that if f'(x) = x^n, then f(x) = x^(n+1)/(n+1) so in this case, f(x) = x^2/2
Andrew
great noticeable direction
Isaac
limit x tend to infinite xcos(π/2x)*sin(π/4x)
Abhijeet Reply
can you give me a problem for function. a trigonometric one
geovanni Reply
Practice Key Terms 1

Get the best Calculus volume 1 course in your pocket!





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