We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Numerical approximations to definite integrals

Study concepts, example questions & explanations for AP Calculus AB

Create An Account

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #6 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

$\begin{align*}&\text{Approximate }\int_{0}^{15}f(x)\\&\text{Using a right Riemann sum, if }f(x)\text{ has the values:}\\&f(0)=-19\\&f(5)=14\\&f(10)=2\\&f(15)=15\end{align*}$

Possible Answers:

180

155

Correct answer:

155

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(0)=-19\\&f(5)=14\\&f(10)=2\\&f(15)=15\\&\text{Although the total interval has a length of }15\\&\text{, notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }5\\&\text{Since we are using the right point of each interval, we disregard the first function value:}\\&\int_{0}^{15}f(x)\approx5[(14)+(2)+(15)]\\&\int_{0}^{15}f(x)\approx155\end{align*}$

Report an Error

Example Question #2 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

$\begin{align*}&\text{Approximate }\int_{-8}^{0}f(x)\\&\text{Using a right Riemann sum, if }f(x)\text{ has the values:}\\&f(-8)=7\\&f(-6)=-7\\&f(-4)=16\\&f(-2)=-16\\&f(0)=20\end{align*}$

Possible Answers:

160

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-8)=7\\&f(-6)=-7\\&f(-4)=16\\&f(-2)=-16\\&f(0)=20\\&\text{Although the total interval has a length of }8\\&\text{, notice the points are equidistant, with }4\text{ intervals between.}\\&\text{Each equidistant interval has length: }2\\&\text{Since we are using the right point of each interval, we disregard the first function value:}\\&\int_{-8}^{0}f(x)\approx2[(-7)+(16)+(-16)+(20)]\\&\int_{-8}^{0}f(x)\approx26\end{align*}$

Report an Error

Example Question #3 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

$\begin{align*}&\text{Approximate }\int_{-6}^{39}f(x)\\&\text{Using a right Riemann sum, if }f(x)\text{ has the values:}\\&f(-6)=-4\\&f(3)=-1\\&f(12)=11\\&f(21)=13\\&f(30)=-16\\&f(39)=-13\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-6)=-4\\&f(3)=-1\\&f(12)=11\\&f(21)=13\\&f(30)=-16\\&f(39)=-13\\&\text{Although the total interval has a length of }45\\&\text{, notice the points are equidistant, with }5\text{ intervals between.}\\&\text{Each equidistant interval has length: }9\\&\text{Since we are using the right point of each interval, we disregard the first function value:}\\&\int_{-6}^{39}f(x)\approx9[(-1)+(11)+(13)+(-16)+(-13)]\\&\int_{-6}^{39}f(x)\approx-54\end{align*}$

Report an Error

Example Question #4 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

$\begin{align*}&\text{Approximate }\int_{-5}^{13}f(x)\\&\text{Using a right Riemann sum, if }f(x)\text{ has the values:}\\&f(-5)=-13\\&f(1)=-12\\&f(7)=17\\&f(13)=7\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-5)=-13\\&f(1)=-12\\&f(7)=17\\&f(13)=7\\&\text{Although the total interval has a length of }18\\&\text{, notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }6\\&\text{Since we are using the right point of each interval, we disregard the first function value:}\\&\int_{-5}^{13}f(x)\approx6[(-12)+(17)+(7)]\\&\int_{-5}^{13}f(x)\approx72\end{align*}$

Report an Error

Example Question #21 : Numerical Approximations To Definite Integrals

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[1,11]:\\&f(1)=10\\&f(3)=3\\&f(5)=-13\\&f(7)=4\\&f(9)=-8\\&f(11)=-15\\&\text{Approximate the integral of }f(x)\text{ on this interval using right Riemann sums.}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(1)=10\\&f(3)=3\\&f(5)=-13\\&f(7)=4\\&f(9)=-8\\&f(11)=-15\\&\text{Although the total interval has a length of }10\\&\text{, notice the points are equidistant, with }5\text{ intervals between.}\\&\text{Each equidistant interval has length: }2\\&\text{Since we are using the right point of each interval, we disregard the first function value:}\\&\int_{1}^{11}f(x)\approx2[(3)+(-13)+(4)+(-8)+(-15)]\\&\int_{1}^{11}f(x)\approx-58\end{align*}$

Report an Error

Example Question #22 : Numerical Approximations To Definite Integrals

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[-7,3]:\\&f(-7)=-18\\&f(-5)=-2\\&f(-3)=-20\\&f(-1)=16\\&f(1)=-12\\&f(3)=-17\\&\text{Approximate the integral of }f(x)\text{ on this interval using right Riemann sums.}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-7)=-18\\&f(-5)=-2\\&f(-3)=-20\\&f(-1)=16\\&f(1)=-12\\&f(3)=-17\\&\text{Although the total interval has a length of }10\\&\text{, notice the points are equidistant, with }5\text{ intervals between.}\\&\text{Each equidistant interval has length: }2\\&\text{Since we are using the right point of each interval, we disregard the first function value:}\\&\int_{-7}^{3}f(x)\approx2[(-2)+(-20)+(16)+(-12)+(-17)]\\&\int_{-7}^{3}f(x)\approx-70\end{align*}$

Report an Error

Example Question #23 : Numerical Approximations To Definite Integrals

$\begin{align*}&\text{Approximate }\int_{5}^{13}f(x)\\&\text{Using a left Riemann sum, if }f(x)\text{ has the values:}\\&f(5)=-7\\&f(7)=-8\\&f(9)=-18\\&f(11)=-8\\&f(13)=-19\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(5)=-7\\&f(7)=-8\\&f(9)=-18\\&f(11)=-8\\&f(13)=-19\\&\text{Although the total interval has a length of }8\\&\text{, notice the points are equidistant, with }4\text{ intervals between.}\\&\text{Each equidistant interval has length: }2\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{5}^{13}f(x)\approx2[(-7)+(-8)+(-18)+(-8)]\\&\int_{5}^{13}f(x)\approx-82\end{align*}$

Report an Error

Example Question #24 : Numerical Approximations To Definite Integrals

$\begin{align*}&\text{Approximate }\int_{-9}^{26}f(x)\\&\text{Using a left Riemann sum, if }f(x)\text{ has the values:}\\&f(-9)=-17\\&f(-2)=11\\&f(5)=17\\&f(12)=1\\&f(19)=-16\\&f(26)=13\end{align*}$

Possible Answers:

315

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-9)=-17\\&f(-2)=11\\&f(5)=17\\&f(12)=1\\&f(19)=-16\\&f(26)=13\\&\text{Although the total interval has a length of }35\\&\text{, notice the points are equidistant, with }5\text{ intervals between.}\\&\text{Each equidistant interval has length: }7\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-9}^{26}f(x)\approx7[(-17)+(11)+(17)+(1)+(-16)]\\&\int_{-9}^{26}f(x)\approx-28\end{align*}$

Report an Error

Example Question #25 : Numerical Approximations To Definite Integrals

$\begin{align*}&f(-10)=-19\\&f(-3)=7\\&f(4)=4\\&f(11)=1\\&\text{Approximate, using a left Riemann sum, }\int_{-10}^{11}f(x)\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-10)=-19\\&f(-3)=7\\&f(4)=4\\&f(11)=1\\&\text{Although the total interval has a length of }21\\&\text{Notice the points are equidistant, with }3\text{ intervals between.}\\&\text{Each equidistant interval has length: }7\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-10}^{11}f(x)\approx7[(-19)+(7)+(4)]\\&\int_{-10}^{11}f(x)\approx-56\end{align*}$

Report an Error

Example Question #26 : Numerical Approximations To Definite Integrals

$\begin{align*}&\text{Approximate }\int_{-6}^{34}f(x)\\&\text{Using a left Riemann sum, if }f(x)\text{ has the values:}\\&f(-6)=8\\&f(2)=2\\&f(10)=-4\\&f(18)=-18\\&f(26)=11\\&f(34)=-7\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A Riemann sum integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx \frac{b-a}{n}(f(x_1)+f(x_2)+...+f(x_n))\\&\text{It is essentially a sum of n rectangles, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable heights :}f(x_i)\text{, which depend on the function value at }x_i\\&\text{In our case, we have function values over an interval:}\\&f(-6)=8\\&f(2)=2\\&f(10)=-4\\&f(18)=-18\\&f(26)=11\\&f(34)=-7\\&\text{Although the total interval has a length of }40\\&\text{Notice the points are equidistant, with }5\text{ intervals between.}\\&\text{Each equidistant interval has length: }8\\&\text{Since we are using the left point of each interval, we disregard the larst function value:}\\&\int_{-6}^{34}f(x)\approx8[(8)+(2)+(-4)+(-18)+(11)]\\&\int_{-6}^{34}f(x)\approx-8\end{align*}$

Report an Error

View AP Calculus AB Tutors

Elan
Certified Tutor

University of Cambridge, Bachelor in Arts, Engineering Physics. University of Illinois at Urbana-Champaign, Doctor of Philoso...

View AP Calculus AB Tutors

Arnav
Certified Tutor

Rutgers University-Newark, Bachelor of Science, Biomedical Engineering.

View AP Calculus AB Tutors

Nafis
Certified Tutor

University of Leipzig, Germany., Bachelor of Science, Physics. University of Pittsburgh-Pittsburgh Campus, Doctor of Philosop...

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in Los Angeles, SAT Tutors in Denver, LSAT Tutors in Denver, Calculus Tutors in Miami, LSAT Tutors in Atlanta, GMAT Tutors in Phoenix, ISEE Tutors in San Francisco-Bay Area, GMAT Tutors in Denver, SAT Tutors in Atlanta, SSAT Tutors in Washington DC

Popular Courses & Classes

GRE Courses & Classes in New York City, ACT Courses & Classes in Phoenix, GMAT Courses & Classes in Seattle, LSAT Courses & Classes in Boston, Spanish Courses & Classes in Miami, GRE Courses & Classes in Atlanta, SSAT Courses & Classes in San Diego, ISEE Courses & Classes in Seattle, SAT Courses & Classes in Chicago, GRE Courses & Classes in Seattle

Popular Test Prep

GMAT Test Prep in Phoenix, MCAT Test Prep in Seattle, LSAT Test Prep in Atlanta, GRE Test Prep in Chicago, GRE Test Prep in New York City, SSAT Test Prep in Denver, GMAT Test Prep in Dallas Fort Worth, SSAT Test Prep in Miami, GMAT Test Prep in Los Angeles, GRE Test Prep in Atlanta

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

AP Calculus AB : Numerical approximations to definite integrals

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #6 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #2 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #3 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #4 : Riemann Sums (Left, Right, And Midpoint Evaluation Points)

Example Question #21 : Numerical Approximations To Definite Integrals

Example Question #22 : Numerical Approximations To Definite Integrals

Example Question #23 : Numerical Approximations To Definite Integrals

Example Question #24 : Numerical Approximations To Definite Integrals

Example Question #25 : Numerical Approximations To Definite Integrals

Example Question #26 : Numerical Approximations To Definite Integrals

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: