We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Integrals

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 #551 : Integrals

Solve the following definite integral.

$\int_0^5 \,2xy+3\,dx$

Possible Answers:

xy^2+3y

x^2y+3x

25y+15

Correct answer:

25y+15

Explanation:

$\int_0^5 \,2xy+3\,dx=x^2y+3x|_{x=0}^{x=5}$

=5^2y+3(5)-[0^2y+3(0)]

=25y+15

Report an Error

Example Question #2302 : Calculus Ii

If $\int_{1}^{7}f(x)dx=5, \int_{3}^{7}f(x)dx=2, \int_{1}^{3}f(x)dx=?$

Possible Answers:

Not enough information.

Correct answer:

Explanation:

If we think of the anti-derivative as computing the area under the curve, then between x=1 and x=7 , the area under our function equals . We also know that between x=3 and x=7 , the area under our function equals . Then, to find the area under the function between x=1 and x=3 , we must simply subtract the other two areas. Therefore:

$\int_{1}^{7}f(x)dx- \int_{3}^{7}f(x)dx=\int_{1}^{3}f(x)dx=5-2=3.$

Report an Error

Example Question #189 : Definite Integrals

If f(x)=x^2-3x+5 , what is $\int_{5}^{5}f(x)dx$ ?

Possible Answers:

$\frac{1}{2}$

$\frac{2}{3}$

Correct answer:

Explanation:

Remember that the anti-derivative computes the area under the curve of the function between the values specified by the upper and lower integral limits. In this question, the upper and lower integral limits match! This, by definition, means that the integral equals There is no area if you start and end at the same point!

Report an Error

Example Question #552 : Integrals

If $\int_{1}^{3} f(x)dx=5, \int_{7}^{3} f(x)dx=-4$ , and $\int_{7}^{10}f(x)dx=1,$ what is $\int_{1}^{10} f(x)dx?$

Possible Answers:

Correct answer:

Explanation:

To find the total integral on our interval from x=1 to x=10 , we need to find the area under each subinterval and add them all up. The first and last part are given in the problem statement, but the middle interval has backwards limits.

Remember:

$\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx.$

Therefore, we need to switch the sign on the middle area. Therefore,

$\int_{1}^{10} f(x)dx=\int_{1}^{3} f(x)dx+-\int_{7}^{3} f(x)dx+\int_{7}^{10} f(x)dx$

=5+--4+1=10.

Report an Error

Example Question #204 : Finding Integrals

In exponentially decaying systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:

$g(t)=\int_{0}^{t}e^{-s(t-b)}*f(b) db$

Where is a constant and f(b) is a function that represents an external force. g(t) represents the overall solution to the differential equation.

Determine g(t) if the external force is given by f(b)=1

Possible Answers:

$g(t)=1-e^{-st}$

$g(t)=\frac{1}{s}[1+e^{-st}]$

$g(t)=\frac{1}{s}[1-e^{-st}]$

$g(t)=-\frac{1}{s}[1-e^{-st}]$

Correct answer:

$g(t)=\frac{1}{s}[1-e^{-st}]$

Explanation:

We can rewrite the integral as

$g(t)=\int_{0}^{t}e^{-s(t-b)}*1 db$

Using substitution, we let

u=-st+sb

$\frac{du}{db}=s$

$db=\frac{du}{s}$

We can therefore write the integral as:

$g(t)=\frac{1}{s}\int_{0}^{t}e^udu$

Since $\int e^u=e^u$ .

$g(t)=\frac{1}{s}[e^u]_{0}^{t}= \frac{1}{s}[e^{(-st+sb)}]_{0}^{t}= \frac{1}{s}[1-e^{-st}]$

Report an Error

Example Question #193 : Definite Integrals

In exponentially decaying systems, often times the solutions to differential equations take on the form of an integral called Duhamel's Integral. This is given by:

$g(t)=\int_{0}^{t}e^{-s(t-b)}*f(b) db$

Where is a constant and f(b) is a function that represents an external force.

Determine the value at t=0 if f(b)=b .

Possible Answers:

$\frac{1}{s^2}$

s^2-1

$\frac{s+1-e^{-s}}{s^2}$

Correct answer:

Explanation:

Plugging f(b) into the equation:

$g(t)=\int_{0}^{t}e^{-s(t-b)}*b db$

Integrating by parts, we can assign values for , , , and :

U=b $dV=e^{-s(t-b)}db$

dU=db $V=\int e^{-s(t-b)}db=\frac{1}{s}e^{-s(t-b)}$

Plugging into our equation:

$\int_{0}^{t}UdV= [UV]_{0}^{t}-\int_{0}^{t}VdU$

$\int_{0}^{t}e^{-s(t-b)}*b db=\frac{1}{s}[e^{-s(t-b)}b]_{0}^{t}-\frac{1}{s}\int_{0}^{t}e^{-s(t-b)}db$

The first term becomes:

$\frac{1}{s}[e^{-s(t-b)}b]_{0}^{t}= \frac{1}{s}[t-0]=\frac{t}{s}$

The integral term becomes $\int_{0}^{t}e^{-s(t-b)}db=\frac{1}{s}[e^{-s(t-b)}]_{0}^{t}=\frac{1}{s}[e^{-s(t-t)}-e^{-st}]=\frac{1-e^{-st}}{s}$

Plugging everything else in we get:

$g(t)=\frac{t}{s}+\frac{e^{-st}-1}{s^2}$

$g(0)=\frac{0}{s}+\frac{e^{0}-1}{s^2}=\frac{0+1-1}{s}=0$

Report an Error

Example Question #553 : Integrals

Evaluate the following definite integral:

$\int^1_{-1}(4x+1)dx$

Possible Answers:

Correct answer:

Explanation:

This integral requires use of the power rule for antiderivatives, which simplifies as follows:

$\int^1_{-1}(4x+1)dx=(2x^2+x)|^1_{-1}=(2(1)^2+1)-(2(-1)^2-1)=5-3=2$

Report an Error

Example Question #192 : Definite Integrals

Evaluate the following definite integral:

$\int^1_0 \frac{x^4}{8-x^5}dx$

Possible Answers:

0.793

0.0561

0.0324

0.0267

0.610

Correct answer:

0.0267

Explanation:

Make the substitution:

u=8-x^5

Where

$du=-5x^4dx \rightarrow x^4dx=-\frac{du}{5}$

The limits of the integral, 0 and 1, are also changed in the substitution:

$0\rightarrow 8-0^5=8$

$1\rightarrow 8-1^5=7$

Using this in the original expression:

$\int^1_0 \frac{x^4}{8-x^5}dx=-\frac{1}{5}\int^7_8\frac{du}{u}= -\frac{1}{5}\cdot \ln(u)|^7_8=-\frac{1}{5}[\ln(7)-\ln(8)]$

$=-\frac{1}{5}\ln(\frac{7}{8})=0.0267$

Report an Error

Example Question #193 : Definite Integrals

Evaluate the following definite integral:

$\int^2_0 xe^{2x}dx$

Possible Answers:

$\frac{e^2+2}{4}$

$e^{4}$

$\frac{e^4}{2}$

6e^4+4

$\frac{3e^4+1}{4}$

Correct answer:

$\frac{3e^4+1}{4}$

Explanation:

To solve this with integration by parts, we rewrite the expression in the form

$\int u \: dv$

where

$u=x, \: \: \:dv=e^{2x}\, dx$

and

$du=dx,\: \: \:v=\frac{e^{2x}}{2}$

To integrate, apply the formula for integration by parts:

$\int u \: dv=uv-\int vdu=\frac{xe^{2x}}{2}|^2_0-\int^2_0\frac{e^{2x}}{2}dx =(\frac{xe^{2x}}{2}-\frac{e^{2x}}{4})|^2_0$

$=\frac{e^{2x}}{4}(2x-1)|^2_0=\frac{e^{4}}{4}(3)-(\frac{1}{4})(-1)=\frac{3e^{4}+1}{4}$

Report an Error

Example Question #194 : Definite Integrals

Evaluate the following definite integral:

$\int^5_4 \frac{10x-24}{x^2-5x+6}\,dx$

Possible Answers:

5.78

6.98

6.24

7.13

2.60

Correct answer:

5.78

Explanation:

This integral can be solved by using partial fractions. First, we have to factor the denominator write the fraction as a sum of two fractions:

$\int^5_4 \frac{10x-24}{x^2-5x+6}\,dx=\int^5_4 \frac{10x-24}{(x-2)(x-3)}\,dx=\int^5_4(\frac{A}{x-2}+\frac{B}{x-3})dx$

Next, we can solve for A and B:

10x-24=A(x-3)+B(x-2)

When we let x=3:

$30-24=0+B\rightarrow B=6$

When x=2:

$20-24=-A+0\rightarrow A=4$

Replacing A and B in the integral, we can now solve it:

$\int^5_4(\frac{4}{x-2}+\frac{6}{x-3})dx=[4\ln(x-2)+6\ln(x-3) ] |^5_4$

$[4\ln(3)+6\ln(2)]-[4\ln(2)+6\ln(1)]=5.78$

Report an Error

View Calculus 2 Tutors

Muhammad
Certified Tutor

Quaid-E-Azam University, Master of Science, Applied Mathematics. Quaid-E-Azam University, Master of Philosophy, Applied Lingu...

View Calculus 2 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Calculus 2 Tutors

John
Certified Tutor

The University of Texas at Arlington, Bachelor of Science, Mathematics. Stanford University, Master of Science, Aerospace Eng...

All Calculus 2 Resources

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

Popular Subjects

English Tutors in Chicago, ACT Tutors in Dallas Fort Worth, Statistics Tutors in Denver, Chemistry Tutors in Miami, Physics Tutors in New York City, SAT Tutors in Phoenix, Math Tutors in Miami, Biology Tutors in Phoenix, SAT Tutors in Boston, Chemistry Tutors in Denver

Popular Courses & Classes

MCAT Courses & Classes in Phoenix, GMAT Courses & Classes in Los Angeles, SAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Houston, Spanish Courses & Classes in Washington DC, MCAT Courses & Classes in Philadelphia, SAT Courses & Classes in Philadelphia, LSAT Courses & Classes in Boston, MCAT Courses & Classes in Washington DC, SAT Courses & Classes in Boston

Popular Test Prep

GMAT Test Prep in Chicago, SSAT Test Prep in Chicago, SAT Test Prep in Washington DC, SSAT Test Prep in New York City, GMAT Test Prep in Atlanta, GMAT Test Prep in San Diego, SSAT Test Prep in Denver, GRE Test Prep in Dallas Fort Worth, MCAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Phoenix

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 : Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #551 : Integrals

Example Question #2302 : Calculus Ii

Example Question #189 : Definite Integrals

Example Question #552 : Integrals

Example Question #204 : Finding Integrals

Example Question #193 : Definite Integrals

Example Question #553 : Integrals

Example Question #192 : Definite Integrals

Example Question #193 : Definite Integrals

Example Question #194 : Definite Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: