Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Double Integrals

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #6 : Double Integration Over General Regions

Evaluate the double integral

$\int_{0}^{3} \int_{0}^{4} 3x+6\,dydx$

Possible Answers:

324

256

126

200

Correct answer:

126

Explanation:

When solving double integrals, we compute the integral on the inside first.

$\int_{0}^{3} \int_{0}^{4} 3x+6\,dydx=\int_0^33xy+6y|_{y=0}^{y=4}\,dx$

$=\int_0^33x(4)+6(4)-[3x(0)+6(0)]\,dx$

$=\int_0^312x+24\,dx$

$=6x^2+24x|_{x=0}^{x=3}$

=6(3)^2+24(3)-6(0)^2-24(0)

=126

Report an Error

Example Question #7 : Double Integration Over General Regions

Evaluate the integral $\int_{1}^{3}\int_{0}^{2}(xy^3)dxdy$

Possible Answers:

Correct answer:

Explanation:

First, you must evaluate the integral with respect to x. This gets you $y^3\frac{x^2}{2}dy$ evaluated from x=0 to x=2 . This becomes $\int_{1}^{3}2y^3dy$ . Solving this integral with respect to y gets you $\frac{1}{2}y^4$ . Evaluating from y=1 to y=3 , you get $\frac{1}{2}(\frac{3^4}{4}-\frac{1}{4})=10$ .

Report an Error

Example Question #11 : Double Integration Over General Regions

Evaluate the following integral: $\int_{0}^{1}\int_{3}^{6}(6x^2z)dzdx$

Possible Answers:

Correct answer:

Explanation:

First, you must evaluate the integral with respect to z. Using the rules for integration, we get 3x^2z^2dx evaluated from z=3 to z=6 . The result is $\int_{0}^{1}54x^2dx$ . This becomes 27x^3 , evaluated from x=0 to x=1 . The final answer is .

Report an Error

Example Question #671 : Multiple Integration

Evaluate:

$\int_{1}^{3}\int_{y}^{2y}dxdy$

Possible Answers:

Correct answer:

Explanation:

To evaluate the iterated integral, we start with the innermost integral, evaluated with respect to x:

$\int_{y}^{2y}dx=x\left.\begin{matrix} 2y\\y \end{matrix}\right|=2y-y=y$

The integral was found using the following rule:

$\int dx=x+C$

Now, we evaluate the last remaining integral, using our answer above from the previous integral as our integrand:

$\int_{1}^{3}ydy=\frac{y^2}{2}\left.\begin{matrix} 3\\1 \end{matrix}\right|=\frac{9}{2}-\frac{1}{2}=4$

The integral was found using the following rule:

$\int x^ndx=\frac{x^{n+1}}{n+1}+C$

Report an Error

Example Question #11 : Double Integration Over General Regions

Evaluate the double integral

$\int_{0}^{1} \int_{0}^{2}x+y\,dx\,dy$

Possible Answers:

Correct answer:

Explanation:

To evaluate the double integral, compute the inside integral first.

$\int_{0}^{1} \int_{0}^{2}x+y\,dx\,dy=\int_0^1 (\frac{1}{2}x^2+xy|_{x=0}^{x=2}dy$

$=\int_0^12+2y\,dy$

$=2y+y^2|_{y=0}^{y=1}$

=2(1)+1^2-[2(0)+0^2]

Report an Error

Example Question #14 : Double Integration Over General Regions

Evaluate the double integral

$\int_0^2 \int_0^1 y-2x\,dx\,dy$

Possible Answers:

Correct answer:

Explanation:

aTo evaluate the double integral, compute the inside integral first.

$\int_{0}^{2} \int_{0}^{1}y-2x\,dx\,dy=\int_0^2 (xy-x^2|_{x=0}^{x=1}dy$

$=\int_0^2 y-1\,dy$

$=\frac{1}{2}y^2-y|_{y=0}^{y=2}$

$=\frac{1}{2}(2)^2-2-[\frac{1}{2}(0)^2-0]$

Report an Error

Example Question #672 : Multiple Integration

Evaluate the double integral

$\int_0^3 \int_0^4 2x-2y \,dy\,dx$

Possible Answers:

Correct answer:

Explanation:

To evaluate the double integral, compute the inside integral first.

$\int_{0}^{3} \int_{0}^{4}2x-2y\,dy\,dx=\int_0^3 2xy-y^2|_{y=0}^{y=4}dx$

$=\int_0^3 8x-16\,dx$

$=4x^2-16x|_{x=0}^{x=3}$

=4(3)^2-16(3)-[4(0)^2-16(0)]

=-12

Report an Error

Example Question #191 : Double Integrals

Integrate:

$\int_{0}^{1} \int_{0}^{5x}4ydydx$

Possible Answers:

$\frac{50}{3}$

$\frac{25}{3}$

Correct answer:

$\frac{50}{3}$

Explanation:

To perform the iterated integration, we must work from inside outwards. To start we perform the following integration:

$\int_{0}^{5x}4ydy=2y^2 \Big|^{5x}_{0}= 2(5x)^2=50x^2$

This becomes the integrand for the outermost integral:

$\int_{0}^{1}50x^2dx=\frac{50x^3}{3}\Big|^{1}_{0}= \frac{50}{3}$

Report an Error

Example Question #192 : Double Integrals

$\int_{0}^{2}\int_{y}^{3y}2xdxdy$

Possible Answers:

$\frac{64}{3}$

$\frac{32}{3}$

$\frac{16}{3}$

Correct answer:

$\frac{64}{3}$

Explanation:

To perform the iterated integral, we work from inside outwards.

The first integral we perform is

$\int_{y}^{3y}2xdx=x^2\left.\begin{matrix} 3y\\y \end{matrix}\right|=9y^2-y^2=8y^2$

This becomes the integrand for the outermost integral,

$\int_{0}^{2}8y^2dy=\frac{8y^3}{3}\left.\begin{matrix} 2\\ 0 \end{matrix}\right|=\frac{64}{3}$

Report an Error

Example Question #13 : Double Integration Over General Regions

Solve:

$\int_{1}^{2}\int_{0}^{x^2}ydydx$

Possible Answers:

$\frac{31}{2}$

$\frac{31}{10}$

$\frac{32}{10}$

Correct answer:

$\frac{31}{10}$

Explanation:

To evaluate the iterated integral, we must work from inside outward.

The first integral we evaluate is

$\int_{0}^{x^2}ydy= \frac{y^2}{2}\left.\begin{matrix} x^2\\0 \end{matrix}\right|= \frac{x^4}{2}$

This becomes the integrand for the outermost integral.

The final integral we evaluate is

$\int_{1}^{2}\frac{x^4}{2}dx= \frac{x^5}{10}\left.\begin{matrix} 2\\1 \end{matrix}\right|= \frac{32}{10}-\frac{1}{10}=\frac{31}{10}$

Report an Error

View Calculus 3 Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

View Calculus 3 Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

View Calculus 3 Tutors

Dawn
Certified Tutor

University of Oregon, Bachelor in Arts, Biochemistry and Molecular Biology. Johns Hopkins University, Doctor of Philosophy, E...

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in San Diego, MCAT Tutors in Seattle, French Tutors in Houston, Math Tutors in Philadelphia, Chemistry Tutors in Denver, English Tutors in Miami, GRE Tutors in San Francisco-Bay Area, Biology Tutors in New York City, Algebra Tutors in Boston, French Tutors in Washington DC

Popular Courses & Classes

ACT Courses & Classes in Philadelphia, MCAT Courses & Classes in Philadelphia, SSAT Courses & Classes in Phoenix, ISEE Courses & Classes in Atlanta, GMAT Courses & Classes in New York City, SSAT Courses & Classes in Denver, Spanish Courses & Classes in Denver, LSAT Courses & Classes in Denver, GRE Courses & Classes in San Diego, ISEE Courses & Classes in San Francisco-Bay Area

Popular Test Prep

ACT Test Prep in Philadelphia, ISEE Test Prep in Philadelphia, SAT Test Prep in Chicago, SSAT Test Prep in Denver, ISEE Test Prep in San Diego, MCAT Test Prep in Philadelphia, ISEE Test Prep in San Francisco-Bay Area, LSAT Test Prep in Denver, ISEE Test Prep in Denver, SAT 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 3 : Double Integrals

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #6 : Double Integration Over General Regions

Example Question #7 : Double Integration Over General Regions

Example Question #11 : Double Integration Over General Regions

Example Question #671 : Multiple Integration

Example Question #11 : Double Integration Over General Regions

Example Question #14 : Double Integration Over General Regions

Example Question #672 : Multiple Integration

Example Question #191 : Double Integrals

Example Question #192 : Double Integrals

Example Question #13 : Double Integration Over General Regions

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: