We are open Saturday and Sunday!

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 #1 : Triple Integration In Cartesian Coordinates

Evaluate the following integral

$\int \int \int_D 15xyz \ dV$ , where $D=[1,2]\times[3,4]\times [5,6]$

Possible Answers:

$\frac{65}{8}$

$\frac{3465}{8}$

$\frac{1}{8}$

Correct answer:

$\frac{3465}{8}$

Explanation:

$\int_{1}^{2} \int_{3}^{4} \int_{5}^{6} 15xyz \ dx\ dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{15}{2}x^2yz\Big|_{5}^{6} dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} (\frac{15}{2}(6)^2yz)-(\frac{15}{2}(5)^2yz) dy\ dz$

$=\int_{1}^{2} \int_{3}^{4} \frac{165}{2}yz\ dy\ dz$

$=\int_{1}^{2} \frac{165}{4}y^2z\Big|_{3}^{4}\ dz$

$=\int_{1}^{2} \frac{165}{4}(4)^2z-\frac{165}{4}(3)^2z\ dz$

$=\int_{1}^{2} \frac{1155}{4}z\ dz$

$=\frac{1155}{8}z^2\Big|_{1}^{2}$

$=\frac{1155}{8}(2)^2-\frac{1155}{8}(1)^2=\frac{3465}{8}$

Report an Error

Example Question #1 : Triple Integration In Cartesian Coordinates

Find the value of the triple integral $\int \int_{B} \int yz^2dxdydz$ , $B=[0,1]\times[0,2]\times[0,3]$

Possible Answers:

Correct answer:

Explanation:

An integral expression of the form

$\int \int_{B} \int f(x,y,z)dxdydz$ , $B=[a,b]\times[c,d]\times[s,t]$

is equivalent to an integral of the form

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz$

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

It is worth noting that the order the integration is performed does not matter, since each variable will be integrated independently, with the others being treated as constants for the purposes of integration. For example

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx$

Considering our integral:

$\int \int_{B} \int yz^2dxdydz$ , $B=[0,1]\times[0,2]\times[0,3]$

The approach is simply to take it step by step:

$\int_{0}^{3}\int_{0}^{2}\int_{0}^{1}yz^2dxdydz=\int_{0}^{3}\int_{0}^{2}xyz^2|_{0}^{1}dydz$

$\int_{0}^{3}\int_{0}^{2}yz^2dydz=\int_{0}^{3}\frac{y^2z^2}{2}|_{0}^{2}dz$

$\int_{0}^{3}2z^2dz=\frac{2z^3}{3}|_{0}^{3}=18$

Report an Error

Example Question #131 : Multiple Integration

Find the value of the triple integral $\int \int_{B} \int (x+y+z)dxdydz$ , $B=[0,1]\times[0,2]\times[0,3]$

Possible Answers:

Correct answer:

Explanation:

An integral expression of the form

$\int \int_{B} \int f(x,y,z)dxdydz$ , $B=[a,b]\times[c,d]\times[s,t]$

is equivalent to an integral of the form

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz$

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx$

Considering our integral:

$\int \int_{B} \int (x+y+z)dxdydz$ , $B=[0,1]\times[0,2]\times[0,3]$

The approach is simply to take it step by step:

$\int_{0}^{3}\int_{0}^{2}\int_{0}^{1}(x+y+z)dxdydz=\int_{0}^{3}\int_{0}^{2}(\frac{x^2}{2}+xy+xz)|_{0}^{1}dydz$

$\int_{0}^{3}\int_{0}^{2}(y+z+\frac{1}{2})dydz=\int_{0}^{3}(\frac{y^2}{2}+yz+\frac{y}{2})|_{0}^{2}dz$

$\int_{0}^{3}(2z+3)dz=(z^2+3z)|_{0}^{3}=18$

Report an Error

Example Question #132 : Multiple Integration

Find the value of the triple integral $\int \int_{B} \int (x^2-yz)dxdydz$ , $B=[0,3]\times[-4,0]\times[-1,1]$

Possible Answers:

Correct answer:

Explanation:

An integral expression of the form

$\int \int_{B} \int f(x,y,z)dxdydz$ , $B=[a,b]\times[c,d]\times[s,t]$

is equivalent to an integral of the form

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz$

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx$

Considering our integral:

$\int \int_{B} \int (x^2-yz)dxdydz$ , $B=[0,3]\times[-4,0]\times[-1,1]$

The approach is simply to take it step by step:

$\int_{-1}^{1}\int_{-4}^{0}\int_{0}^{3}(x^2-yz)dxdydz=\int_{-1}^{1}\int_{-4}^{0}(\frac{x^3}{3}-xyz)|_{0}^{3}dydz$

$\int_{-1}^{1}\int_{-4}^{0}(9-3yz)dydz=\int_{-1}^{1}(9y-\frac{3y^2z}{2})|_{-4}^{0}dz$

$\int_{-1}^{1}(36+24z)dz=(36z+12z^2)|_{-1}^{1}=72$

Report an Error

Example Question #1 : Triple Integration In Cartesian Coordinates

Find the value of the triple integral $\int \int_{B} \int (3x^2z^3e^{xyz^2})dxdydz$ , $B=[0,1]\times[0,2]\times[0,3]$

Possible Answers:

$\frac{e^{18}-179}{72}$

$\frac{e^{18}-179}{24}$

$\frac{e^{18}-181}{24}$

$\frac{e^{18}-181}{72}$

$\frac{e^{18}+179}{72}$

Correct answer:

$\frac{e^{18}-181}{24}$

Explanation:

An integral expression of the form

$\int \int_{B} \int f(x,y,z)dxdydz$ , $B=[a,b]\times[c,d]\times[s,t]$

is equivalent to an integral of the form

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz$

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx$

Considering our integral:

$\int \int_{B} \int (3x^2z^3e^{xyz^2})dxdydz$ , $B=[0,1]\times[0,2]\times[0,3]$

The approach is simply to take it step by step. But first, we should consider the order in which we wish to integrate; those and terms in front of the exponent are daunting. Let's see if we can eliminate some by integrating with respect to initially:

${\int_{0}^{3}\int_{0}^{1}\int_{0}^{2}(3x^2z^3e^{xyz^2})dydxdz=\int_{0}^{3}\int_{0}^{1}(3xze^{xyz^2})|_{0}^{2}dxdz=\int_{0}^{3}\int_{0}^{1}(3xze^{2xz^2}-3xz)dxdz}$

This is already looking more manageable. Now, however, we have both an and in front of our exponent, and we must decide which one to choose for the next integration. Recall that $\frac{d}{du}e^{u^2}=2ue^{u^2}$ . Integrating with respect to would be a better way of eliminating the terms in front of the exponent:

$\int_{0}^{1}\int_{0}^{3}(3xze^{2xz^2}-3xz)dzdx=\int_{0}^{1}(\frac{3}{4}e^{2xz^2}-\frac{3}{2}xz^2)|_{0}^{3}dx=\frac{3}{4}e^{18x}-\frac{27}{2}x-\frac{3}{4}$

Finally, we're left with a (relatively) simpler integral:

$\int_{0}^{1}(\frac{3}{4}e^{18x}-\frac{27}{2}x-\frac{3}{4})dx=(\frac{e^{18x}}{24}-\frac{27}{4}x^2-\frac{3}{4}x)|_{0}^{1}=\frac{e^{18}-181}{24}$

Being careful about the order of integration can make the work much easier (relatively).

Report an Error

Example Question #2 : Triple Integration In Cartesian Coordinates

Find the value of the triple integral $\int \int_{B} \int (-xy^2sin(xyz))dxdydz$ , $B=[0,\pi]\times[0,\frac{1}{2}]\times[0,1]$

Possible Answers:

$\frac{8+\pi^2}{8}$

$\frac{8\pi+\pi^2}{8}$

$\frac{8-\pi^2}{8\pi}$

$\frac{8-\pi}{8\pi}$

$\frac{8-\pi^2}{8}$

Correct answer:

$\frac{8-\pi^2}{8\pi}$

Explanation:

An integral expression of the form

$\int \int_{B} \int f(x,y,z)dxdydz$ , $B=[a,b]\times[c,d]\times[s,t]$

is equivalent to an integral of the form

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz$

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx$

Considering our integral:

$\int \int_{B} \int (-xy^2sin(xyz))dxdydz$ , $B=[0,\pi]\times[0,\frac{1}{2}]\times[0,1]$

The approach is simply to take it step by step; however, to make the job of integration easier, it might be prudent to begin by integrating with respect to :

$\int_{0}^{\pi}\int_{0}^{\frac{1}{2}}\int_{0}^{1}(-xy^2sin(xyz))dzdydx=\int_{0}^{\pi}\int_{0}^{\frac{1}{2}}ycos(xyz)|_{0}^{1}dydx$

That worked out pretty well in terms of getting rid of some terms in the front; let's use that trick again and take the integral with respect to next:

$\int_{0}^{\frac{1}{2}}\int_{0}^{\pi}(ycos(xy)-y)dxdy=\int_{0}^{\frac{1}{2}}(sin(xy)-xy)|_{0}^{\pi}dy$

This leaves a relatively straightforward integral with respect to :

$\int_{0}^{\frac{1}{2}}(sin(\pi y)-\pi y)dy=(\frac{-cos(\pi y)}{\pi}-\frac{\pi y^2}{2})|_{0}^{\frac{1}{2}}=\frac{8-\pi^2}{8\pi}$

Being careful about the order of integration can make a complicated problem a bit simpler.

Report an Error

Example Question #134 : Multiple Integration

Find the value of the triple integral $\int \int_{B} \int xydxdydz$ , $B=[0,3]\times[1,2]\times[-1,3]$

Possible Answers:

$\frac{83}{4}$

Correct answer:

Explanation:

An integral expression of the form

$\int \int_{B} \int f(x,y,z)dxdydz$ , $B=[a,b]\times[c,d]\times[s,t]$

is equivalent to an integral of the form

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz$

Where the innermost integral corresponds to the innermost derivative variable, and the outermost integral corresponds to the outermost variable.

$\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx$

Considering our integral:

$\int \int_{B} \int xydxdydz$ , $B=[0,3]\times[1,2]\times[-1,3]$

The approach is simply to take it step by step:

$\int_{-1}^{3}\int_{1}^{2}\int_{0}^{3}xydxdydz=\int_{-1}^{3}\int_{1}^{2}\frac{x^2y}{2}|_{0}^{3}dydz$

$\int_{-1}^{3}\int_{1}^{2}\frac{9y}{2}dydz=\int_{-1}^{3}\frac{9y^2}{4}|_{1}^{2}dz$

$\int_{-1}^{3}dz=\frac{27}{4}z|_{-1}^{3}=27$

Report an Error

Example Question #2 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-13}^{2}\int_{-13}^{-7}\int_{6}^{9}(3x + y + 2z)dxdydz\end{align*}$

Possible Answers:

405

Correct answer:

405

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-13}^{2}\int_{-13}^{-7}\int_{6}^{9}(3x + y + 2z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-13}^{2}\int_{-13}^{-7}\int_{6}^{9}(3x + y + 2z)dxdydz=\int_{-13}^{2}\int_{-13}^{-7}(x{(y + 2z)} + {(3x^2)}/2)dydz|_{6}^{9}\\&\int_{-13}^{2}\int_{-13}^{-7}(3y + 6z + 135/2)dydz=\int_{-13}^{2}({(3y{(y + 4z + 45)})}/2)dz|_{-13}^{-7}\\&\int_{-13}^{2}(36z + 225)dz=9z{(2z + 25)}dz|_{-13}^{2}=405\end{align*}$

Report an Error

Example Question #3 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{8}^{20}\int_{-7}^{-2}\int_{-11}^{18}(6xy - 2z)dxdydz\end{align*}$

Possible Answers:

527220

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{8}^{20}\int_{-7}^{-2}\int_{-11}^{18}(6xy - 2z)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{8}^{20}\int_{-7}^{-2}\int_{-11}^{18}(6xy - 2z)dxdydz=\int_{8}^{20}\int_{-7}^{-2}(3x^2y - 2xz)dydz|_{-11}^{18}\\&\int_{8}^{20}\int_{-7}^{-2}(609y - 58z)dydz=\int_{8}^{20}({(29y{(21y - 4z)})}/2)dz|_{-7}^{-2}\\&\int_{8}^{20}(- 290z - 27405/2)dz=-{(145z{(2z + 189)})}/2dz|_{8}^{20}=-213150\end{align*}$

Report an Error

Example Question #142 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-4}^{3}\int_{-8}^{-2}\int_{4}^{6}(24xyz^2)dxdydz\end{align*}$

Possible Answers:

127680

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-4}^{3}\int_{-8}^{-2}\int_{4}^{6}(24xyz^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-4}^{3}\int_{-8}^{-2}\int_{4}^{6}(24xyz^2)dxdydz=\int_{-4}^{3}\int_{-8}^{-2}(12x^2yz^2)dydz|_{4}^{6}\\&\int_{-4}^{3}\int_{-8}^{-2}(240yz^2)dydz=\int_{-4}^{3}(120y^2z^2)dz|_{-8}^{-2}\\&\int_{-4}^{3}(-7200z^2)dz=-2400z^3dz|_{-4}^{3}=-218400\end{align*}$

Report an Error

View Calculus 3 Tutors

Asad
Certified Tutor

NED University of Engineering and Technology, Bachelor of Science, Electrical Engineering. University of Cincinnati-Main Camp...

View Calculus 3 Tutors

Mohammed
Certified Tutor

University of Maryland-College Park, Bachelor of Science, Electrical Engineering. University of Maryland-College Park, Master...

View Calculus 3 Tutors

Therar
Certified Tutor

Beirut Arab University, Doctor of Philosophy, Mathematics. Saint Joseph University of Beirut, Master of Science, Counselor Ed...

All Calculus 3 Resources

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

Popular Subjects

Algebra Tutors in Phoenix, SSAT Tutors in Chicago, Math Tutors in San Diego, Math Tutors in Washington DC, Chemistry Tutors in Atlanta, English Tutors in San Francisco-Bay Area, Reading Tutors in Denver, Algebra Tutors in Denver, Spanish Tutors in Los Angeles, ACT Tutors in Phoenix

Popular Courses & Classes

ISEE Courses & Classes in Philadelphia, LSAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Phoenix, GRE Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Denver, Spanish Courses & Classes in Los Angeles, SSAT Courses & Classes in New York City, MCAT Courses & Classes in Houston, GRE Courses & Classes in Chicago, SSAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

SSAT Test Prep in Chicago, SSAT Test Prep in Washington DC, GMAT Test Prep in Miami, GMAT Test Prep in Denver, ACT Test Prep in Phoenix, MCAT Test Prep in Chicago, SSAT Test Prep in Atlanta, MCAT Test Prep in Seattle, SSAT Test Prep in Dallas Fort Worth, GRE Test Prep in Boston

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 #1 : Triple Integration In Cartesian Coordinates

Example Question #1 : Triple Integration In Cartesian Coordinates

Example Question #131 : Multiple Integration

Example Question #132 : Multiple Integration

Example Question #1 : Triple Integration In Cartesian Coordinates

Example Question #2 : Triple Integration In Cartesian Coordinates

Example Question #134 : Multiple Integration

Example Question #2 : Triple Integration In Cartesian Coordinates

Example Question #3 : Triple Integration In Cartesian Coordinates

Example Question #142 : Multiple Integration

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: