Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #21 : Riemann Sums

$\begin{align*}&\text{Calculate the Riemann sums integral approximation of :}\\&\int_{4}^{6.4}(11ln(x))dx\\&\text{Using left points over }4\text{ intervals.}\end{align*}$

Possible Answers:

408.70

41.70

19.86

254.40

Correct answer:

41.70

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{4}^{6.4}(11ln(x))dx\\&\text{So the interval is}[4,6.4]\text{ the subintervals have length}\frac{6.4-(4)}{4}=0.6\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[4,\frac{23}{5},\frac{26}{5},\frac{29}{5}]\\&\int_{4}^{6.4}(11ln(x))dx=0.6[(11ln(4))+(11ln(\frac{23}{5}))+(11ln(\frac{26}{5}))+(11ln(\frac{29}{5}))]\\&\int_{4}^{6.4}(11ln(x))dx=41.70\end{align*}$

Report an Error

Example Question #22 : Riemann Sums

$\begin{align*}&\text{Using the method of left point Riemann sums approximate the integral:}\\&\int_{5}^{7.1}(20tan(3x))dx\\&\text{Using }3\text{ intervals.}\end{align*}$

Possible Answers:

374.26

70.62

649.66

190.66

Correct answer:

70.62

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{5}^{7.1}(20tan(3x))dx\\&\text{So the interval is}[5,7.1]\text{ the subintervals have length}\frac{7.1-(5)}{3}=0.7\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[5,\frac{57}{10},\frac{32}{5}]\\&\int_{5}^{7.1}(20tan(3x))dx=0.7[(20tan(15))+(20tan(\frac{171}{10}))+(20tan(\frac{96}{5}))]\\&\int_{5}^{7.1}(20tan(3x))dx=70.62\end{align*}$

Report an Error

Example Question #23 : Riemann Sums

$\begin{align*}&\text{Using }3\text{ intervals, and the method of left point Riemann sums approximate the integral:}\\&\int_{1}^{12.4}(-14x^{2})dx\end{align*}$

Possible Answers:

-10948.56

-766.71

-51614.64

-5213.60

Correct answer:

-5213.60

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{1}^{12.4}(-14x^{2})dx\\&\text{So the interval is}[1,12.4]\text{ the subintervals have length}\frac{12.4-(1)}{3}=3.8\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[1,\frac{24}{5},\frac{43}{5}]\\&\int_{1}^{12.4}(-14x^{2})dx=3.8[(-14)+(-\frac{8064}{25})+(-\frac{25886}{25})]\\&\int_{1}^{12.4}(-14x^{2})dx=-5213.60\end{align*}$

Report an Error

Example Question #24 : Riemann Sums

$\begin{align*}&\text{Using }3\text{ intervals, and the method of left point Riemann sums approximate the integral:}\\&\int_{-1}^{6.8}(-3tan(3x))dx\end{align*}$

Possible Answers:

314.74

10.79

13.88

87.43

Correct answer:

87.43

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{-1}^{6.8}(-3tan(3x))dx\\&\text{So the interval is}[-1,6.8]\text{ the subintervals have length}\frac{6.8-(-1)}{3}=2.6\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[-1,\frac{8}{5},\frac{21}{5}]\\&\int_{-1}^{6.8}(-3tan(3x))dx=2.6[(3tan(3))+(-3tan(\frac{24}{5}))+(-3tan(\frac{63}{5}))]\\&\int_{-1}^{6.8}(-3tan(3x))dx=87.43\end{align*}$

Report an Error

Example Question #25 : Riemann Sums

$\begin{align*}&\text{Using }4\text{ intervals, and the method of left point Riemann sums approximate the integral:}\\&\int_{2}^{20.8}(15tan(x))dx\end{align*}$

Possible Answers:

-44.55

-568.47

-258.40

-27.20

Correct answer:

-258.40

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{2}^{20.8}(15tan(x))dx\\&\text{So the interval is}[2,20.8]\text{ the subintervals have length}\frac{20.8-(2)}{4}=4.7\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[2,\frac{67}{10},\frac{57}{5},\frac{161}{10}]\\&\int_{2}^{20.8}(15tan(x))dx=4.7[(15tan(2))+(15tan(\frac{67}{10}))+(15tan(\frac{57}{5}))+(15tan(\frac{161}{10}))]\\&\int_{2}^{20.8}(15tan(x))dx=-258.40\end{align*}$

Report an Error

Example Question #26 : Riemann Sums

$\begin{align*}&\text{Calculate the Riemann sums integral approximation of :}\\&\int_{3}^{11.1}(3tan(2x))dx\\&\text{Using left points over }3\text{ intervals.}\end{align*}$

Possible Answers:

-0.57

-34.05

-2.84

-5.68

Correct answer:

-5.68

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{3}^{11.1}(3tan(2x))dx\\&\text{So the interval is}[3,11.1]\text{ the subintervals have length}\frac{11.1-(3)}{3}=2.7\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[3,\frac{57}{10},\frac{42}{5}]\\&\int_{3}^{11.1}(3tan(2x))dx=2.7[(3tan(6))+(3tan(\frac{57}{5}))+(3tan(\frac{84}{5}))]\\&\int_{3}^{11.1}(3tan(2x))dx=-5.68\end{align*}$

Report an Error

Example Question #27 : Riemann Sums

$\begin{align*}&\text{Calculate the Riemann sums integral approximation of :}\\&\int_{0}^{3.6}(-3cos(6x))dx\\&\text{Using left points over }4\text{ intervals.}\end{align*}$

Possible Answers:

-0.18

-1.51

-8.60

-0.52

Correct answer:

-1.51

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{0}^{3.6}(-3cos(6x))dx\\&\text{So the interval is}[0,3.6]\text{ the subintervals have length}\frac{3.6-(0)}{4}=0.9\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[0,\frac{9}{10},\frac{9}{5},\frac{27}{10}]\\&\int_{0}^{3.6}(-3cos(6x))dx=0.9[(-3)+(-3cos(\frac{27}{5}))+(-3cos(\frac{54}{5}))+(-3cos(\frac{81}{5}))]\\&\int_{0}^{3.6}(-3cos(6x))dx=-1.51\end{align*}$

Report an Error

Example Question #28 : Riemann Sums

$\begin{align*}&\text{Using }4\text{ intervals, and the method of left point Riemann sums approximate the integral:}\\&\int_{4}^{6.8}(3ln(6x))dx\end{align*}$

Possible Answers:

251.25

28.55

7.32

171.30

Correct answer:

28.55

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{4}^{6.8}(3ln(6x))dx\\&\text{So the interval is}[4,6.8]\text{ the subintervals have length}\frac{6.8-(4)}{4}=0.7\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[4,\frac{47}{10},\frac{27}{5},\frac{61}{10}]\\&\int_{4}^{6.8}(3ln(6x))dx=0.7[(3ln(24))+(3ln(\frac{141}{5}))+(3ln(\frac{162}{5}))+(3ln(\frac{183}{5}))]\\&\int_{4}^{6.8}(3ln(6x))dx=28.55\end{align*}$

Report an Error

Example Question #79 : Integrals

$\begin{align*}&\text{Using the method of left point Riemann sums approximate the integral:}\\&\int_{3}^{15.9}(13ln(4x))dx\\&\text{Using }3\text{ intervals.}\end{align*}$

Possible Answers:

258.11

542.03

3252.16

4498.83

Correct answer:

542.03

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{3}^{15.9}(13ln(4x))dx\\&\text{So the interval is}[3,15.9]\text{ the subintervals have length}\frac{15.9-(3)}{3}=4.3\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[3,\frac{73}{10},\frac{58}{5}]\\&\int_{3}^{15.9}(13ln(4x))dx=4.3[(13ln(12))+(13ln(\frac{146}{5}))+(13ln(\frac{232}{5}))]\\&\int_{3}^{15.9}(13ln(4x))dx=542.03\end{align*}$

Report an Error

Example Question #71 : Introduction To Integrals

$\begin{align*}&\text{Calculate the Riemann sums integral approximation of :}\\&\int_{4}^{21.6}(16sin(4x))dx\\&\text{Using left points over }4\text{ intervals.}\end{align*}$

Possible Answers:

411.04

627.38

19.49

72.11

Correct answer:

72.11

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{We're asked to approximate}\int_{4}^{21.6}(16sin(4x))dx\\&\text{So the interval is}[4,21.6]\text{ the subintervals have length}\frac{21.6-(4)}{4}=4.4\\&\text{and since we are using the *left*point of each interval, the x-values are:}\\&[4,\frac{42}{5},\frac{64}{5},\frac{86}{5}]\\&\int_{4}^{21.6}(16sin(4x))dx=4.4[(16sin(16))+(16sin(\frac{168}{5}))+(16sin(\frac{256}{5}))+(16sin(\frac{344}{5}))]\\&\int_{4}^{21.6}(16sin(4x))dx=72.11\end{align*}$

Report an Error

View Calculus 2 Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

View Calculus 2 Tutors

Fatemeh
Certified Tutor

University of Isfahan, Doctor of Engineering, Chemical Engineering. University of Windsor, Doctor of Engineering, Environment...

View Calculus 2 Tutors

MEER
Certified Tutor

IIT , Master's/Graduate, Microelectronics .

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Chemistry Tutors in Washington DC, Chemistry Tutors in Denver, LSAT Tutors in San Francisco-Bay Area, Statistics Tutors in Dallas Fort Worth, French Tutors in Washington DC, Physics Tutors in Philadelphia, Spanish Tutors in Houston, ISEE Tutors in San Francisco-Bay Area, Calculus Tutors in Denver, SSAT Tutors in Phoenix

Popular Courses & Classes

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

Popular Test Prep

LSAT Test Prep in Atlanta, SAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Atlanta, MCAT Test Prep in Denver, LSAT Test Prep in Phoenix, GMAT Test Prep in Seattle, GRE Test Prep in Houston, GMAT Test Prep in Houston, SAT Test Prep in Chicago, GRE Test Prep in San Diego

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 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #21 : Riemann Sums

Example Question #22 : Riemann Sums

Example Question #23 : Riemann Sums

Example Question #24 : Riemann Sums

Example Question #25 : Riemann Sums

Example Question #26 : Riemann Sums

Example Question #27 : Riemann Sums

Example Question #28 : Riemann Sums

Example Question #79 : Integrals

Example Question #71 : Introduction To Integrals

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: