Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Calculus 3

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 #184 : Multiple Integration

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

Possible Answers:

$8e^{(-5/2)}{(e^{(1/2)} - 1)} - e^{(4)} + 1$

$2e^{(2)} + 4e^{(5)} - 4e^{(6)} - 2$

$2e^{(2)} - 4e^{(-4)} + 4e^{(-5)} - 2$

$4e^{(5)} - 4e^{(6)} + 8e^{(-5/2)}{(e^{(1/2)} - 1)}$

$8e^{(5/2)}{(e^{(1/2)} - 1)} - e^{(4)} + 1$

Correct answer:

$2e^{(2)} + 4e^{(5)} - 4e^{(6)} - 2$

Explanation:

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

Report an Error

Example Question #185 : Multiple Integration

$\begin{align*}&\text{Evaluate the triple integral}\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x + sin{(y/2)})dxdydz\end{align*}$

Possible Answers:

10cos{(1)} - 10cos{(3)} + 10

10cos{(1)} - 10cos{(3)} - 10

40cos{(1/2)} + 40

40cos{(1/2)} - 50

8cos{(3/2)} - 8cos{(1)} + 80

Correct answer:

40cos{(1/2)} + 40

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x + sin{(y/2)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x + sin{(y/2)})dxdydz=\int_{-3}^{2}\int_{-1}^{0}({(x{(x + 2sin{(y/2)})})}/2)dydz|_{2}^{6}\\&\int_{-3}^{2}\int_{-1}^{0}(4sin{(y/2)} + 16)dydz=\int_{-3}^{2}(16y - 8cos{(y/2)})dz|_{-1}^{0}\\&\int_{-3}^{2}(8cos{(1/2)} + 8)dz=z{(8cos{(1/2)} + 8)}dz|_{-3}^{2}=40cos{(1/2)} + 40\end{align*}$

Report an Error

Example Question #181 : Triple Integrals

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

Possible Answers:

15cos{(5)} - 15285/4

25cos{(2)} - 25cos{(5)} + 9375/4

15cos{(5)} - 3885/4

25cos{(2)} - 25cos{(5)} - 3825/4

15cos{(1)} - 15cos{(4)} + 9375/4

Correct answer:

25cos{(2)} - 25cos{(5)} + 9375/4

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-5}^{-2}\int_{0}^{5}\int_{-4}^{1}(y^3 - sin{(z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-5}^{-2}\int_{0}^{5}\int_{-4}^{1}(y^3 - sin{(z)})dxdydz=\int_{-5}^{-2}\int_{0}^{5}(-x{(sin{(z)} - y^3)})dydz|_{-4}^{1}\\&\int_{-5}^{-2}\int_{0}^{5}(5y^3 - 5sin{(z)})dydz=\int_{-5}^{-2}({(5y^4)}/4 - 5ysin{(z)})dz|_{0}^{5}\\&\int_{-5}^{-2}(3125/4 - 25sin{(z)})dz={(3125z)}/4 + 25cos{(z)}dz|_{-5}^{-2}=25cos{(2)} - 25cos{(5)} + 9375/4\end{align*}$

Report an Error

Example Question #51 : Triple Integration In Cartesian Coordinates

$\begin{align*}&\text{Evaluate the triple integral}\int_{-2}^{1}\int_{5}^{7}\int_{9}^{10}(5z^9)dxdydz\end{align*}$

Possible Answers:

409064436

19539646797

1023

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{1}\int_{5}^{7}\int_{9}^{10}(5z^9)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{1}\int_{5}^{7}\int_{9}^{10}(5z^9)dxdydz=\int_{-2}^{1}\int_{5}^{7}(5xz^9)dydz|_{9}^{10}\\&\int_{-2}^{1}\int_{5}^{7}(5z^9)dydz=\int_{-2}^{1}(5yz^9)dz|_{5}^{7}\\&\int_{-2}^{1}(10z^9)dz=z^{10}dz|_{-2}^{1}=-1023\end{align*}$

Report an Error

Example Question #188 : Multiple Integration

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

Possible Answers:

48700

10900

107500

220900

157900

Correct answer:

48700

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{1}^{2}\int_{-4}^{1}\int_{3}^{8}(12{(3x + 4y)}^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{1}^{2}\int_{-4}^{1}\int_{3}^{8}(12{(3x + 4y)}^2)dxdydz=\int_{1}^{2}\int_{-4}^{1}({(4{(3x + 4y)}^3)}/3)dydz\Big|_{3}^{8}\\&\int_{1}^{2}\int_{-4}^{1}(7920y + 960y^2 + 17460)dydz=\int_{1}^{2}(20y{(198y + 16y^2 + 873)})\Big|_{-4}^{1}\\&\int_{1}^{2}(48700)dz=48700zdz\Big|_{1}^{2}=48700\end{align*}$

Report an Error

Example Question #189 : Multiple Integration

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

Possible Answers:

18.52

-9.07

-117.87

43.02

Correct answer:

18.52

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x^3e^{(y - 4)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-3}^{2}\int_{-1}^{0}\int_{2}^{6}(x^3e^{(y - 4)})dxdydz=\int_{-3}^{2}\int_{-1}^{0}(0.004579x^4e^{(y)})dydz|_{2}^{6}\\&\int_{-3}^{2}\int_{-1}^{0}(5.861e^{(y)})dydz=\int_{-3}^{2}(5.861e^{(y)})dz|_{-1}^{0}\\&\int_{-3}^{2}(3.7049)dz=3.705zdz|_{-3}^{2}=18.524\end{align*}$

Report an Error

Example Question #190 : Multiple Integration

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

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-1}^{4}\int_{-3}^{2}\int_{-5}^{-3}(4x + 2y)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-1}^{4}\int_{-3}^{2}\int_{-5}^{-3}(4x + 2y)dxdydz=\int_{-1}^{4}\int_{-3}^{2}(2x{(x + y)})dydz|_{-5}^{-3}\\&\int_{-1}^{4}\int_{-3}^{2}(4y - 32)dydz=\int_{-1}^{4}(2y{(y - 16)})dz|_{-3}^{2}\\&\int_{-1}^{4}(-170)dz=-170zdz|_{-1}^{4}=-850\end{align*}$

Report an Error

Example Question #191 : Triple Integrals

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

Possible Answers:

396

120

Correct answer:

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{-2}^{1}\int_{4}^{8}\int_{1}^{2}(6x - 2y)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{-2}^{1}\int_{4}^{8}\int_{1}^{2}(6x - 2y)dxdydz=\int_{-2}^{1}\int_{4}^{8}(x{(3x - 2y)})dydz|_{1}^{2}\\&\int_{-2}^{1}\int_{4}^{8}(9 - 2y)dydz=\int_{-2}^{1}(-y{(y - 9)})dz|_{4}^{8}\\&\int_{-2}^{1}(-12)dz=-12zdz|_{-2}^{1}=-36\end{align*}$

Report an Error

Example Question #192 : Triple Integrals

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

Possible Answers:

-182.63

-196.63

29.68

17.68

Correct answer:

-196.63

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{4}^{5}\int_{-3}^{-1}\int_{1}^{2}(x + 3y - e^{(z)})dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{4}^{5}\int_{-3}^{-1}\int_{1}^{2}(x + 3y - e^{(z)})dxdydz=\int_{4}^{5}\int_{-3}^{-1}(x{(3y - e^{(z)})} + x^2/2)dydz|_{1}^{2}\\&\int_{4}^{5}\int_{-3}^{-1}(3y - e^{(z)} + 3/2)dydz=\int_{4}^{5}({(3y^2)}/2 - y{(e^{(z)} - 3/2)})dz|_{-3}^{-1}\\&\int_{4}^{5}(- 2e^{(z)} - 9)dz=- 9z - 2e^{(z)}dz|_{4}^{5}=-196.63\end{align*}$

Report an Error

Example Question #191 : Triple Integrals

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

Possible Answers:

700.34

628.79

892.79

808.34

Correct answer:

700.34

Explanation:

$\begin{align*}&\text{When performing a triple integral, it is worth noting that the order}\\&\text{in which the integration performed does not entirely matter, since each variable will be}\\&\text{integrated independently, with the others being treated as constants for the}\\&\text{puposes of integration. For example:}\\&\int_{s}^{t} \int_{c}^{d} \int_{a}^b f(x,y,z)dxdydz=\int_{a}^{b} \int_{c}^{d} \int_{s}^t f(x,y,z)dzdydx\\&\text{Considering our integral:}\\&\int_{2}^{5}\int_{2}^{6}\int_{-5}^{-4}(e^{(z - 2)} + 3y^2)dxdydz\\&\text{The approach is simply to take it step by step:}\\&\int_{2}^{5}\int_{2}^{6}\int_{-5}^{-4}(e^{(z - 2)} + 3y^2)dxdydz=\int_{2}^{5}\int_{2}^{6}(x{(e^{(z - 2)} + 3y^2)})dydz|_{-5}^{-4}\\&\int_{2}^{5}\int_{2}^{6}(e^{(z - 2)} + 3y^2)dydz=\int_{2}^{5}(y^3 + ye^{(-2)}e^{(z)})dz|_{2}^{6}\\&\int_{2}^{5}(4e^{(-2)}e^{(z)} + 208)dz=208z + 4e^{(-2)}e^{(z)}dz|_{2}^{5}=700.34\end{align*}$

Report an Error

View Calculus 3 Tutors

Madeleine
Certified Tutor

Emory University, Bachelor of Science, Engineering Science.

View Calculus 3 Tutors

Manasa
Certified Tutor

Ramaiah Institute of Technology, Bachelor of Engineering, Telecommunications Engineering.

View Calculus 3 Tutors

Ioannis
Certified Tutor

Elmhurst University, Bachelor of Science, Physics.

All Calculus 3 Resources

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

Popular Subjects

Chemistry Tutors in Philadelphia, ACT Tutors in San Diego, Algebra Tutors in Atlanta, Chemistry Tutors in Seattle, Biology Tutors in Miami, SAT Tutors in Seattle, English Tutors in Houston, LSAT Tutors in San Francisco-Bay Area, Algebra Tutors in Washington DC, ISEE Tutors in Boston

Popular Courses & Classes

LSAT Courses & Classes in San Diego, GMAT Courses & Classes in Phoenix, GMAT Courses & Classes in Miami, ISEE Courses & Classes in New York City, MCAT Courses & Classes in Phoenix, ACT Courses & Classes in Miami, Spanish Courses & Classes in Houston, LSAT Courses & Classes in Denver, GRE Courses & Classes in Philadelphia, GMAT Courses & Classes in Seattle

Popular Test Prep

ACT Test Prep in New York City, ISEE Test Prep in Phoenix, SAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in Seattle, ISEE Test Prep in Denver, ISEE Test Prep in New York City, LSAT Test Prep in Boston, GMAT Test Prep in Seattle, ACT Test Prep in Miami, GRE Test Prep in Los Angeles

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 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #184 : Multiple Integration

Example Question #185 : Multiple Integration

Example Question #181 : Triple Integrals

Example Question #51 : Triple Integration In Cartesian Coordinates

Example Question #188 : Multiple Integration

Example Question #189 : Multiple Integration

Example Question #190 : Multiple Integration

Example Question #191 : Triple Integrals

Example Question #192 : Triple Integrals

Example Question #191 : Triple Integrals

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: