We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Triple 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 #161 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-8}^{-3}\int_{3}^{4}\int_{1}^{5}(2z^3 - 4y)dxdydz\end{align*}$

Possible Answers:

1510

2190

1280

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-8}^{-3}\int_{3}^{4}\int_{1}^{5}(2z^3 - 4y)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-8}^{-3}\int_{3}^{4}\int_{1}^{5}(2z^3 - 4y)dxdydz=\int_{-8}^{-3}\int_{3}^{4}(-x{(4y - 2z^3)})dydz|_{1}^{5}\\&\int_{-8}^{-3}\int_{3}^{4}(8z^3 - 16y)dydz=\int_{-8}^{-3}(-8y{(y - z^3)})dz|_{3}^{4}\\&\int_{-8}^{-3}(8z^3 - 56)dz=2z{(z^3 - 28)}dz|_{-8}^{-3}=-8310\end{align*}$

Report an Error

Example Question #28 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{5}\int_{1}^{6}\int_{-7}^{-6}(24{(y + 2z)}^2 + 3x^2)dxdydz\end{align*}$

Possible Answers:

1015

1345

19675

11425

17515

Correct answer:

19675

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{5}\int_{1}^{6}\int_{-7}^{-6}(24{(y + 2z)}^2 + 3x^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5}\int_{1}^{6}\int_{-7}^{-6}(24{(y + 2z)}^2 + 3x^2)dxdydz=\int_{4}^{5}\int_{1}^{6}(x^3 + 24x{(y + 2z)}^2)dydz|_{-7}^{-6}\\&\int_{4}^{5}\int_{1}^{6}(24{(y + 2z)}^2 + 127)dydz=\int_{4}^{5}(127y + 8{(y + 2z)}^3)dz|_{1}^{6}\\&\int_{4}^{5}(1680z + 480z^2 + 2355)dz=5z{(168z + 32z^2 + 471)}dz|_{4}^{5}=19675\end{align*}$

Report an Error

Example Question #29 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{7}^{11}\int_{10}^{11}\int_{-3}^{0}(8xz^3)dxdydz\end{align*}$

Possible Answers:

771120

1002456

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{7}^{11}\int_{10}^{11}\int_{-3}^{0}(8xz^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{7}^{11}\int_{10}^{11}\int_{-3}^{0}(8xz^3)dxdydz=\int_{7}^{11}\int_{10}^{11}(4x^2z^3)dydz|_{-3}^{0}\\&\int_{7}^{11}\int_{10}^{11}(-36z^3)dydz=\int_{7}^{11}(-36yz^3)dz|_{10}^{11}\\&\int_{7}^{11}(-36z^3)dz=-9z^4dz|_{7}^{11}=-110160\end{align*}$

Report an Error

Example Question #31 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{5}^{10}\int_{-2}^{3}\int_{1}^{5}(2x + 4y - 3x^2)dxdydz\end{align*}$

Possible Answers:

500

600

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{10}\int_{-2}^{3}\int_{1}^{5}(2x + 4y - 3x^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{10}\int_{-2}^{3}\int_{1}^{5}(2x + 4y - 3x^2)dxdydz=\int_{5}^{10}\int_{-2}^{3}(x{(x + 4y - x^2)})dydz|_{1}^{5}\\&\int_{5}^{10}\int_{-2}^{3}(16y - 100)dydz=\int_{5}^{10}(4y{(2y - 25)})dz|_{-2}^{3}\\&\int_{5}^{10}(-460)dz=-460zdz|_{5}^{10}=-2300\end{align*}$

Report an Error

Example Question #32 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{3}^{6}\int_{-4}^{0}\int_{0}^{2}(6x - 2y + 3y^2)dxdydz\end{align*}$

Possible Answers:

1440

624

1128

696

Correct answer:

624

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{3}^{6}\int_{-4}^{0}\int_{0}^{2}(6x - 2y + 3y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{3}^{6}\int_{-4}^{0}\int_{0}^{2}(6x - 2y + 3y^2)dxdydz=\int_{3}^{6}\int_{-4}^{0}(3x^2 - x{(2y - 3y^2)})dydz|_{0}^{2}\\&\int_{3}^{6}\int_{-4}^{0}(6y^2 - 4y + 12)dydz=\int_{3}^{6}(2y{(y^2 - y + 6)})dz|_{-4}^{0}\\&\int_{3}^{6}(208)dz=208zdz|_{3}^{6}=624\end{align*}$

Report an Error

Example Question #33 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{2}\int_{-4}^{-2}\int_{-5}^{-3}(5x + 2y - z)dxdydz\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{2}\int_{-4}^{-2}\int_{-5}^{-3}(5x + 2y - z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{2}\int_{-4}^{-2}\int_{-5}^{-3}(5x + 2y - z)dxdydz=\int_{-1}^{2}\int_{-4}^{-2}({(5x^2)}/2 + x{(2y - z)})dydz|_{-5}^{-3}\\&\int_{-1}^{2}\int_{-4}^{-2}(4y - 2z - 40)dydz=\int_{-1}^{2}(-2y{(z - y + 20)})dz|_{-4}^{-2}\\&\int_{-1}^{2}(- 4z - 104)dz=-2z{(z + 52)}dz|_{-1}^{2}=-318\end{align*}$

Report an Error

Example Question #34 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{7}\int_{0}^{5}\int_{0}^{5}(2x + 6y^2 + 14)dxdydz\end{align*}$

Possible Answers:

7045

19125

9125

8625

Correct answer:

8625

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{7}\int_{0}^{5}\int_{0}^{5}(2x + 6y^2 + 14)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{7}\int_{0}^{5}\int_{0}^{5}(2x + 6y^2 + 14)dxdydz=\int_{2}^{7}\int_{0}^{5}(x{(6y^2 + 14)} + x^2)dydz|_{0}^{5}\\&\int_{2}^{7}\int_{0}^{5}(30y^2 + 95)dydz=\int_{2}^{7}(95y + 10y^3)dz|_{0}^{5}\\&\int_{2}^{7}(1725)dz=1725zdz|_{2}^{7}=8625\end{align*}$

Report an Error

Example Question #35 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{3}\int_{-2}^{3}\int_{-3}^{1}(14x - 3y^2)dxdydz\end{align*}$

Possible Answers:

560

740

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{3}\int_{-2}^{3}\int_{-3}^{1}(14x - 3y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{3}\int_{-2}^{3}\int_{-3}^{1}(14x - 3y^2)dxdydz=\int_{2}^{3}\int_{-2}^{3}(x{(7x - 3y^2)})dydz|_{-3}^{1}\\&\int_{2}^{3}\int_{-2}^{3}(- 12y^2 - 56)dydz=\int_{2}^{3}(-4y{(y^2 + 14)})dz|_{-2}^{3}\\&\int_{2}^{3}(-420)dz=-420zdz|_{2}^{3}=-420\end{align*}$

Report an Error

Example Question #161 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{3}\int_{2}^{7}\int_{-3}^{-1}(6y + e^{(x)})dxdydz\end{align*}$

Possible Answers:

$10e^{(3)} - 10e^{(-2)} + 1350$

$25e^{(-1)} - 25e^{(-3)} + 150$

$10e^{(3)} - 10e^{(-2)} - 600$

$25e^{(-1)} - 25e^{(-3)} + 1350$

$10e^{(7)} - 10e^{(2)} - 600$

Correct answer:

$25e^{(-1)} - 25e^{(-3)} + 1350$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{3}\int_{2}^{7}\int_{-3}^{-1}(6y + e^{(x)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{3}\int_{2}^{7}\int_{-3}^{-1}(6y + e^{(x)})dxdydz=\int_{-2}^{3}\int_{2}^{7}(e^{(x)} + 6xy)dydz|_{-3}^{-1}\\&\int_{-2}^{3}\int_{2}^{7}(12y + e^{(-1)} - e^{(-3)})dydz=\int_{-2}^{3}(y{(e^{(-1)} - e^{(-3)})} + 6y^2)dz|_{2}^{7}\\&\int_{-2}^{3}(5e^{(-1)} - 5e^{(-3)} + 270)dz=z{(5e^{(-1)} - 5e^{(-3)} + 270)}dz|_{-2}^{3}=25e^{(-1)} - 25e^{(-3)} + 1350\end{align*}$

Report an Error

Example Question #37 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{5}^{6}\int_{1}^{3}\int_{2}^{3}(ycos{(x)} + 3z^2)dxdydz\end{align*}$

Possible Answers:

{(5sin{(3)})}/2 - {(5sin{(1)})}/2 + 182

4sin{(3)} - 4sin{(2)} + 182

5sin{(6)} - 5sin{(5)} + 26

11sin{(3)} - 11sin{(2)} + 26

4sin{(6)} - 4sin{(5)} + 38

Correct answer:

4sin{(3)} - 4sin{(2)} + 182

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{5}^{6}\int_{1}^{3}\int_{2}^{3}(ycos{(x)} + 3z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{5}^{6}\int_{1}^{3}\int_{2}^{3}(ycos{(x)} + 3z^2)dxdydz=\int_{5}^{6}\int_{1}^{3}(3xz^2 + ysin{(x)})dydz|_{2}^{3}\\&\int_{5}^{6}\int_{1}^{3}(3z^2 - y{(sin{(2)} - sin{(3)})})dydz=\int_{5}^{6}(3yz^2 - {(y^2{(sin{(2)} - sin{(3)})})}/2)dz|_{1}^{3}\\&\int_{5}^{6}(4sin{(3)} - 4sin{(2)} + 6z^2)dz=2z^3 - z{(4sin{(2)} - 4sin{(3)})}dz|_{5}^{6}=4sin{(3)} - 4sin{(2)} + 182\end{align*}$

Report an Error

View Calculus 3 Tutors

Amir
Certified Tutor

University of Mississippi, Master of Science, Structural Engineering. University of South Florida-Main Campus, Doctor of Scie...

View Calculus 3 Tutors

Reza
Certified Tutor

University of Sherbrooke, Doctor of Philosophy, Mathematics. University of Manitoba, Master of Science, Mathematics.

View Calculus 3 Tutors

Robert Edward
Certified Tutor

University of Calgary, Master of Engineering, Chemistry. University of Ottawa, Bachelor of Engineering, Civil Engineering.

All Calculus 3 Resources

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

Popular Subjects

ACT Tutors in San Francisco-Bay Area, Spanish Tutors in Denver, Math Tutors in Washington DC, ISEE Tutors in Dallas Fort Worth, French Tutors in Houston, Reading Tutors in Dallas Fort Worth, Reading Tutors in Atlanta, Reading Tutors in Philadelphia, GRE Tutors in San Diego, ACT Tutors in Los Angeles

Popular Courses & Classes

GMAT Courses & Classes in Denver, SSAT Courses & Classes in Denver, SSAT Courses & Classes in Seattle, MCAT Courses & Classes in Philadelphia, ISEE Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Denver, Spanish Courses & Classes in Los Angeles, SSAT Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Los Angeles, Spanish Courses & Classes in San Diego

Popular Test Prep

MCAT Test Prep in San Diego, GRE Test Prep in Philadelphia, ISEE Test Prep in Los Angeles, SAT Test Prep in Washington DC, SAT Test Prep in San Francisco-Bay Area, ACT Test Prep in Miami, GRE Test Prep in Denver, LSAT Test Prep in Los Angeles, SAT Test Prep in San Diego, GMAT Test Prep in New York City

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 : Triple Integrals

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #161 : Multiple Integration

Example Question #28 : Triple Integration In Cartesian Coordinates

Example Question #29 : Triple Integration In Cartesian Coordinates

Example Question #31 : Triple Integration In Cartesian Coordinates

Example Question #32 : Triple Integration In Cartesian Coordinates

Example Question #33 : Triple Integration In Cartesian Coordinates

Example Question #34 : Triple Integration In Cartesian Coordinates

Example Question #35 : Triple Integration In Cartesian Coordinates

Example Question #161 : Triple Integrals

Example Question #37 : Triple Integration In Cartesian Coordinates

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: