Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Integral Applications

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Application Of Integrals

Find the work done by gravity exerting an acceleration of $-10\frac{m}{s^2}$ for a 10kg block down from its original position with no initial velocity.

Remember that

$F_{grav}=mass*acceleration$

$W=\int_{a}^{b}F(x)dx$ , where F(x) is a force measured in Newtons , is work measured in Joules , and and are initial and final positions respectively.

Possible Answers:

250J

-500J

1000J

100J

Correct answer:

250J

Explanation:

The force of gravity is proportional to the mass of the object and acceleration of the object.

$F=ma= (10*-10)\frac{kg*m}{s^2}= -100N$

Since the block fell down 5 meters, its final position is and initial position is .

$\\ W=\int_{0}^{-5}-100dx\\ \\=\frac{-100x}{2}|_{0}^{-5} \\ \\=-50x|_{0}^{-5}\\ \\=-50(-5)--50(0)\\ \\=250J$

Report an Error

Example Question #1931 : Calculus Ii

The velocity of a car is defined by the equation $v(t)=4x^{2}-3x$ , where is the time in minutes. What distance (in meters) does the car travel between t=0 and t=4 ?

Possible Answers:

600m

60 m

$\frac{3}{184} m$

$\frac{184}{3} m$

$\frac{190}{3}m$

Correct answer:

$\frac{184}{3} m$

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or $\int_{a}^{b}v(t)dt$ .

Since $v(t)=4x^{2}-3x$ , we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{0}^{4}(4x^{2}-3x)dx$

$=\left[\frac{4}{3}x^{3}-\frac{3}{2}x^{2}\right]_{0}^{4}\textrm{}$

$=\left[\frac{4}{3}(4)^{3}-\frac{3}{2}(4)^{2}\right]-\left[\frac{4}{3}(0)^{3}-\frac{3}{2}(0)^{2}\right]$

$=\left[\frac{256}{3}-\frac{48}{2}\right]$

$=\frac{512}{6}-\frac{144}{6}$

$=\frac{368}{6}$

$=\frac{184}{3}$

Report an Error

Example Question #11 : Integral Applications

The velocity of a train is defined by the equation $v(t)=x^{2}+6x-5$ , where is the time in seconds What distance (in meters) does the train travel between t=0 and t=3 ?

Possible Answers:

20m

21 m

10m

12m

25m

Correct answer:

21 m

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or $\int_{a}^{b}v(t)dt$ .

Since $v(t)=x^{2}+6x-5$ , we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{0}^{3}(x^{2}+6x-5)dx$

$=\left[\frac{1}{3}x^{3}+3x^{2}-5x\right]_{0}^{3}\textrm{}$

$=\left[\frac{1}{3}(3)^{3}+3(3)^{2}-5(3)\right]-\left[\frac{1}{3}(0)^{3}+3(0)^{2}-5(0)\right]$

$=\frac{1}{3}(3)^{3}+3(3)^{2}-5(3)$

$=\frac{27}{3}+27-15$

=9+27-15

=36-15

=21

Report an Error

Example Question #13 : Integral Applications

The velocity of a balloon is defined by the equation $v(t)=6x^{2}-4x+12$ , where is the time in minutes. What distance (in meters) does the balloon travel between t=0 and t=2 ?

Possible Answers:

34m

32m

30m

23m

33m

Correct answer:

32m

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or $\int_{a}^{b}v(t)dt$ .

Since $v(t)=6x^{2}-4x+12$ , we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{0}^{2}(6x^{2}-4x+12)dx$

$=[2x^{3}-2x^{2}+12x]_{0}^{2}\textrm{}$

$=[2(2)^{3}-2(2)^{2}+12(2)]-[2(0)^{3}-2x=(0)^{2}+12(0)]$

=16-8+24

=8+24

=32

Report an Error

Example Question #14 : Integral Applications

A frisbee has a velocity defined by v(t)=2t+13 , where we express in seconds. What distance does it travel between $0\leq t\leq 2$ in meters?

Possible Answers:

30m

32m

34m

28m

26m

Correct answer:

30m

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or

$\int_{a}^{b}v(t)dt$ .

Since

v(t)=2t+13 ,

we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{0}^{2}(2t+13)dt$

$=[t^{2}+13t]_{0}^{2}\textrm{}$

$=[(2)^{2}+13(2)]-[(0)^{2}+(0)t]$

=4+26

=30

Report an Error

Example Question #15 : Integral Applications

A ball has a velocity defined by v(t)=3t-4 , where we express in seconds. What distance does it travel between $0\leq t\leq 3$ in meters?

Possible Answers:

$\frac{5}{2}m$

$\frac{5}{3}m$

$\frac{2}{3}m$

None of the above

$\frac{3}{2}m$

Correct answer:

$\frac{3}{2}m$

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or $\int_{a}^{b}v(t)dt$ .

Since

v(t)=3t-4 , we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{0}^{3}(3t-4)dt$

$=[\frac{3}{2}t^{2}-4t]_{0}^{3}$

$=[\frac{3}{2}(3)^{2}-4(3)]-[\frac{3}{2}(0)^{2}-4(0)]$

$=\frac{27}{2}-12$

$=\frac{27}{2}-\frac{24}{2}$

$=\frac{3}{2}$

Report an Error

Example Question #11 : Applications In Physics

A subway has a velocity defined by v(t)=11t+9 , where we express in seconds. What distance does it travel between $0\leq t\leq 4$ in meters?

Possible Answers:

128m

126m

124m

130m

122m

Correct answer:

124m

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or $\int_{a}^{b}v(t)dt$ .

Since

v(t)=11t+9 ,

we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{0}^{4}(11t+9)dt$

$=[\frac{11}{2}t^{2}+9t]_{0}^{4}\textrm{}$

$=\frac{176}{2}+36$

=88+36

=124

Report an Error

Example Question #17 : Integral Applications

A train goes a certain distance between $0\leq t \leq 2$ (where is time in seconds). If we know that the train's velocity is defined as v(t)=12t+15 , what is the distance it travelled (in meters)?

Possible Answers:

56m

50m

45m

46m

54m

Correct answer:

54m

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or $\int_{a}^{b}v(t)dt$ .

Since v(t)=12t+15 , we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{0}^{2}(12t+15)dt$

$=[6t^{2}+15t]_{0}^{2}\textrm{}$

$=[6(2)^{2}+15(2)]-[6(0)^{2}+15(0)]$

=[24+30]

=54

Report an Error

Example Question #18 : Integral Applications

A car goes a certain distance between $0\leq t \leq 3$ (where is time in seconds). If we know that the car's velocity is defined as v(t)=14t-2 , what is the distance it travelled (in meters)?

Possible Answers:

55m

59m

58m

56m

57m

Correct answer:

57m

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or $\int_{a}^{b}v(t)dt$ .

Since v(t)=14t-2 , we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{0}^{3}(14t-2)dt$

$=[7t^{2}-2t]_{0}^{3}\textrm{}$

$=[7(3)^{2}-2(3)]-[7(0)^{2}-2(0)]$

=[63-6]

=57

Report an Error

Example Question #11 : Integral Applications

A plane goes a certain distance between $2\leq t \leq 5$ (where is time in seconds). If we know that the plane's velocity is defined as v(t)=6t-2 , what is the distance it travelled (in meters)?

Possible Answers:

60m

61m

57m

58m

59m

Correct answer:

57m

Explanation:

We define velocity as the derivative of distance, or v(t)=d'(t) .

Thus, in order to find the distance traveled when given the velocity, we'll need to take the definite integral of the velocity function over a specified time period, or $\int_{a}^{b}v(t)dt$ .

Since v(t)=6t-2 , we can use the Power Rule for Integrals

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ ,

to find:

$\int_{2}^{5}(6t-2)dt$

$=[3t^{2}-2t]_{2}^{5}\textrm{}$

$=[3(5)^{2}-2(5)]-[3(2)^{2}-2(2)]$

=75-10-12+4

=57

Report an Error

View Calculus 2 Tutors

Chelsea
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Physics. The University of Texas at Dallas, Master of Science, Bioinf...

View Calculus 2 Tutors

Sumit
Certified Tutor

Calcutta University, Bachelor of Science, Mathematics. Indian Statistical Institute, Master of Science, Mathematics.

View Calculus 2 Tutors

Jason
Certified Tutor

Brigham Young University, Bachelor, Neuroscience.

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SSAT Tutors in New York City, French Tutors in San Francisco-Bay Area, Math Tutors in Dallas Fort Worth, Spanish Tutors in Phoenix, English Tutors in Denver, Biology Tutors in Chicago, ACT Tutors in San Francisco-Bay Area, Chemistry Tutors in Miami, MCAT Tutors in Chicago, Spanish Tutors in Atlanta

Popular Courses & Classes

LSAT Courses & Classes in Washington DC, SAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Los Angeles, SAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in San Diego, ACT Courses & Classes in Seattle, SSAT Courses & Classes in Denver, Spanish Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Houston, SAT Courses & Classes in Los Angeles

Popular Test Prep

LSAT Test Prep in Atlanta, GMAT Test Prep in Dallas Fort Worth, SAT Test Prep in Houston, SSAT Test Prep in Houston, GMAT Test Prep in Houston, GRE Test Prep in Washington DC, ACT Test Prep in Houston, ACT Test Prep in Washington DC, SAT Test Prep in Washington DC, ISEE Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 2 : Integral Applications

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Application Of Integrals

Example Question #1931 : Calculus Ii

Example Question #11 : Integral Applications

Example Question #13 : Integral Applications

Example Question #14 : Integral Applications

Example Question #15 : Integral Applications

Example Question #11 : Applications In Physics

Example Question #17 : Integral Applications

Example Question #18 : Integral Applications

Example Question #11 : Integral Applications

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: