Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Triple Integration in Cartesian Coordinates

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 #11 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-10}^{11}\int_{-5}^{9}\int_{0}^{2}(24xy^2 - 8z^3)dxdydz\end{align*}$

Possible Answers:

23520

222264

27048

489216

134064

Correct answer:

27048

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-10}^{11}\int_{-5}^{9}\int_{0}^{2}(24xy^2 - 8z^3)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-10}^{11}\int_{-5}^{9}\int_{0}^{2}(24xy^2 - 8z^3)dxdydz=\int_{-10}^{11}\int_{-5}^{9}(12x^2y^2 - 8xz^3)dydz|_{0}^{2}\\&\int_{-10}^{11}\int_{-5}^{9}(48y^2 - 16z^3)dydz=\int_{-10}^{11}(16y{(y^2 - z^3)})dz|_{-5}^{9}\\&\int_{-10}^{11}(13664 - 224z^3)dz=-56z{(z^3 - 244)}dz|_{-10}^{11}=27048\end{align*}$

Report an Error

Example Question #12 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-1}^{4}\int_{2}^{4}\int_{5}^{8}(12x - 8z + 3y^2)dxdydz\end{align*}$

Possible Answers:

4590

2820

2010

3690

Correct answer:

2820

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{4}\int_{2}^{4}\int_{5}^{8}(12x - 8z + 3y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{4}\int_{2}^{4}\int_{5}^{8}(12x - 8z + 3y^2)dxdydz=\int_{-1}^{4}\int_{2}^{4}(6x^2 - x{(8z - 3y^2)})dydz|_{5}^{8}\\&\int_{-1}^{4}\int_{2}^{4}(9y^2 - 24z + 234)dydz=\int_{-1}^{4}(3y^3 - y{(24z - 234)})dz|_{2}^{4}\\&\int_{-1}^{4}(636 - 48z)dz=-12z{(2z - 53)}dz|_{-1}^{4}=2820\end{align*}$

Report an Error

Example Question #621 : Calculus 3

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

Possible Answers:

226

170

226

170

Correct answer:

170

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{5}\int_{-5}^{-3}\int_{6}^{7}(2x - 4yz)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5}\int_{-5}^{-3}\int_{6}^{7}(2x - 4yz)dxdydz=\int_{4}^{5}\int_{-5}^{-3}(x^2 - 4xyz)dydz|_{6}^{7}\\&\int_{4}^{5}\int_{-5}^{-3}(13 - 4yz)dydz=\int_{4}^{5}(-y{(2yz - 13)})dz|_{-5}^{-3}\\&\int_{4}^{5}(32z + 26)dz=2z{(8z + 13)}dz|_{4}^{5}=170\end{align*}$

Report an Error

Example Question #14 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{6}^{10}\int_{3}^{6}\int_{5}^{6}(4x - 12y)dxdydz\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{6}^{10}\int_{3}^{6}\int_{5}^{6}(4x - 12y)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{6}^{10}\int_{3}^{6}\int_{5}^{6}(4x - 12y)dxdydz=\int_{6}^{10}\int_{3}^{6}(2x{(x - 6y)})dydz|_{5}^{6}\\&\int_{6}^{10}\int_{3}^{6}(22 - 12y)dydz=\int_{6}^{10}(-2y{(3y - 11)})dz|_{3}^{6}\\&\int_{6}^{10}(-96)dz=-96zdz|_{6}^{10}=-384\end{align*}$

Report an Error

Example Question #15 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-6}^{-5}\int_{-3}^{1}\int_{8}^{13}(2x + 4y - 6z)dxdydz\end{align*}$

Possible Answers:

740

1460

100

1000

Correct answer:

1000

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-6}^{-5}\int_{-3}^{1}\int_{8}^{13}(2x + 4y - 6z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-6}^{-5}\int_{-3}^{1}\int_{8}^{13}(2x + 4y - 6z)dxdydz=\int_{-6}^{-5}\int_{-3}^{1}(x^2 + x{(4y - 6z)})dydz|_{8}^{13}\\&\int_{-6}^{-5}\int_{-3}^{1}(20y - 30z + 105)dydz=\int_{-6}^{-5}(10y^2 - y{(30z - 105)})dz|_{-3}^{1}\\&\int_{-6}^{-5}(340 - 120z)dz=-20z{(3z - 17)}dz|_{-6}^{-5}=1000\end{align*}$

Report an Error

Example Question #16 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{2}^{6}\int_{0}^{5}\int_{5}^{8}(5x^4 - 12y^5)dxdydz\end{align*}$

Possible Answers:

217860

Correct answer:

217860

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{6}\int_{0}^{5}\int_{5}^{8}(5x^4 - 12y^5)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{6}\int_{0}^{5}\int_{5}^{8}(5x^4 - 12y^5)dxdydz=\int_{2}^{6}\int_{0}^{5}(x^5 - 12xy^5)dydz|_{5}^{8}\\&\int_{2}^{6}\int_{0}^{5}(29643 - 36y^5)dydz=\int_{2}^{6}(29643y - 6y^6)dz|_{0}^{5}\\&\int_{2}^{6}(54465)dz=54465zdz|_{2}^{6}=217860\end{align*}$

Report an Error

Example Question #141 : Triple Integrals

$\begin{align*}&\text{Evaluate the triple integral}\int_{-5}^{0}\int_{2}^{3}\int_{8}^{13}(6z^2)dxdydz\end{align*}$

Possible Answers:

16850

350

1250

1750

950

Correct answer:

1250

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-5}^{0}\int_{2}^{3}\int_{8}^{13}(6z^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{0}\int_{2}^{3}\int_{8}^{13}(6z^2)dxdydz=\int_{-5}^{0}\int_{2}^{3}(6xz^2)dydz|_{8}^{13}\\&\int_{-5}^{0}\int_{2}^{3}(30z^2)dydz=\int_{-5}^{0}(30yz^2)dz|_{2}^{3}\\&\int_{-5}^{0}(30z^2)dz=10z^3dz|_{-5}^{0}=1250\end{align*}$

Report an Error

Example Question #621 : Calculus 3

$\begin{align*}&\text{Evaluate the triple integral}\int_{4}^{7}\int_{-7}^{-2}\int_{-10}^{-7}(6z^2 - 1/y^2)dxdydz\end{align*}$

Possible Answers:

$\frac{117171}{14}$

$\frac{168795}{14}$

$\frac{84411}{14}$

$\frac{275895}{14}$

$\frac{117135}{14}$

Correct answer:

$\frac{117135}{14}$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{7}\int_{-7}^{-2}\int_{-10}^{-7}(6z^2 - 1/y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{7}\int_{-7}^{-2}\int_{-10}^{-7}(6z^2 - 1/y^2)dxdydz=\int_{4}^{7}\int_{-7}^{-2}(-x{(1/y^2 - 6z^2)})dydz|_{-10}^{-7}\\&\int_{4}^{7}\int_{-7}^{-2}(18z^2 - 3/y^2)dydz=\int_{4}^{7}(18yz^2 + 3/y)dz|_{-7}^{-2}\\&\int_{4}^{7}(90z^2 - 15/14)dz=30z^3 - {(15z)}/14dz|_{4}^{7}=117135/14\end{align*}$

Report an Error

Example Question #151 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{13}\int_{10}^{11}\int_{-2}^{0}(e^{(x + y)})dxdydz\end{align*}$

Possible Answers:

$2{(e^{(8)} - e^{(13)})}{(e^{(10)} - e^{(11)})}$

$3{(e^{(8)} - e^{(13)})}{(e^{(-2)} - 1)}$

$8e^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)}$

$5e^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)}$

${(e^{(8)} - e^{(13)})}{(e^{(-2)} - 1)}$

Correct answer:

$5e^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)}$

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{8}^{13}\int_{10}^{11}\int_{-2}^{0}(e^{(x + y)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{13}\int_{10}^{11}\int_{-2}^{0}(e^{(x + y)})dxdydz=\int_{8}^{13}\int_{10}^{11}(e^{(x + y)})dydz|_{-2}^{0}\\&\int_{8}^{13}\int_{10}^{11}(e^{(-2)}e^{(y)}{(e^{(2)} - 1)})dydz=\int_{8}^{13}(e^{(-2)}e^{(y)}{(e^{(2)} - 1)})dz|_{10}^{11}\\&\int_{8}^{13}(e^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)})dz=ze^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)}dz|_{8}^{13}=5e^{(8)}{(e^{(1)} - 1)}^2{(e^{(1)} + 1)}\end{align*}$

Report an Error

Example Question #11 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-9}^{-6}\int_{4}^{5}\int_{-1}^{0}(12z^3 - 2x)dxdydz\end{align*}$

Possible Answers:

3366

3324

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-9}^{-6}\int_{4}^{5}\int_{-1}^{0}(12z^3 - 2x)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-9}^{-6}\int_{4}^{5}\int_{-1}^{0}(12z^3 - 2x)dxdydz=\int_{-9}^{-6}\int_{4}^{5}(12xz^3 - x^2)dydz|_{-1}^{0}\\&\int_{-9}^{-6}\int_{4}^{5}(12z^3 + 1)dydz=\int_{-9}^{-6}(y{(12z^3 + 1)})dz|_{4}^{5}\\&\int_{-9}^{-6}(12z^3 + 1)dz=z + 3z^4dz|_{-9}^{-6}=-15792\end{align*}$

Report an Error

View Calculus 3 Tutors

Philipp
Certified Tutor

University of Nevada-Las Vegas, Bachelor of Science, Mechanical Engineering.

View Calculus 3 Tutors

Peter
Certified Tutor

Kansas State University, Bachelors, Mathematics. Bowling Green State University-Main Campus, Master of Science, Management In...

View Calculus 3 Tutors

Reuben
Certified Tutor

University of Alberta, Bachelor of Science, Mathematics.

All Calculus 3 Resources

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

Popular Subjects

SSAT Tutors in San Diego, Calculus Tutors in Chicago, Calculus Tutors in Atlanta, LSAT Tutors in Miami, GRE Tutors in Philadelphia, ISEE Tutors in Los Angeles, GMAT Tutors in Atlanta, SSAT Tutors in Philadelphia, SSAT Tutors in Los Angeles, Computer Science Tutors in Denver

Popular Courses & Classes

GRE Courses & Classes in Miami, ISEE Courses & Classes in San Diego, LSAT Courses & Classes in Chicago, SSAT Courses & Classes in Miami, SAT Courses & Classes in Atlanta, ACT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in New York City, MCAT Courses & Classes in Phoenix, GRE Courses & Classes in Dallas Fort Worth, ISEE Courses & Classes in Phoenix

Popular Test Prep

ACT Test Prep in New York City, SSAT Test Prep in Denver, ACT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Miami, SSAT Test Prep in Boston, GRE Test Prep in Dallas Fort Worth, LSAT Test Prep in Atlanta, ISEE Test Prep in Dallas Fort Worth, MCAT Test Prep in Los Angeles, GMAT Test Prep in Houston

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 Integration in Cartesian Coordinates

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #11 : Triple Integration In Cartesian Coordinates

Example Question #12 : Triple Integration In Cartesian Coordinates

Example Question #621 : Calculus 3

Example Question #14 : Triple Integration In Cartesian Coordinates

Example Question #15 : Triple Integration In Cartesian Coordinates

Example Question #16 : Triple Integration In Cartesian Coordinates

Example Question #141 : Triple Integrals

Example Question #621 : Calculus 3

Example Question #151 : Multiple Integration

Example Question #11 : 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: