Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Trapezoidal sums

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 #31 : Trapezoidal Sums

$\begin{align*}&\text{Approximate }\int_{-5}^{17}f(x)\\&\text{Using the trapezoidal rule, if }f(x)\text{ has the values:}\\&f(-5)=5\\&f(1)=-6\\&f(11)=11\\&f(17)=-4\end{align*}$

Possible Answers:

40.3333

121

5.5

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(-5)=5\\&f(1)=-6\\&f(11)=11\\&f(17)=-4\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }6,10,6\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{-5}^{17}f(x)\approx(6)\frac{5+(-6)}{2}+(10)\frac{-6+(11)}{2}+(6)\frac{11+(-4)}{2}\\&\int_{-5}^{17}f(x)\approx43\end{align*}$

Report an Error

Example Question #32 : Trapezoidal Sums

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[1,32]:\\&f(1)=15\\&f(3)=-9\\&f(7)=9\\&f(13)=-15\\&f(17)=14\\&\text{Approximate the integral of }f(x)\text{ on this interval using the trapezoidal rule.}\end{align*}$

Possible Answers:

-3.875

-0.74519

-23.101

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(1)=15\\&f(3)=-9\\&f(7)=9\\&f(13)=-15\\&f(17)=14\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }2,4,6,4\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{1}^{32}f(x)\approx(2)\frac{15+(-9)}{2}+(4)\frac{-9+(9)}{2}+(6)\frac{9+(-15)}{2}+(4)\frac{-15+(14)}{2}\\&\int_{1}^{32}f(x)\approx-14\end{align*}$

Report an Error

Example Question #1106 : Ap Calculus Ab

$\begin{align*}&f(-5)=-2\\&f(-1)=19\\&f(9)=-19\\&f(15)=19\\&f(25)=-13\\&\text{Approximate, using a trapezoidal sum, }\int_{-5}^{25}f(x)\end{align*}$

Possible Answers:

86.25

11.5

345

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(-5)=-2\\&f(-1)=19\\&f(9)=-19\\&f(15)=19\\&f(25)=-13\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }4,10,6,10\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{-5}^{25}f(x)\approx(4)\frac{-2+(19)}{2}+(10)\frac{19+(-19)}{2}+(6)\frac{-19+(19)}{2}+(10)\frac{19+(-13)}{2}\\&\int_{-5}^{25}f(x)\approx64\end{align*}$

Report an Error

Example Question #1107 : Ap Calculus Ab

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[-10,20]:\\&f(-10)=-17\\&f(-2)=19\\&f(2)=6\\&f(10)=-11\\&f(20)=-4\\&\text{Approximate the integral of }f(x)\text{ on this interval using the trapezoidal rule.}\end{align*}$

Possible Answers:

3.5

26.25

105

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(-10)=-17\\&f(-2)=19\\&f(2)=6\\&f(10)=-11\\&f(20)=-4\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }8,4,8,10\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{-10}^{20}f(x)\approx(8)\frac{-17+(19)}{2}+(4)\frac{19+(6)}{2}+(8)\frac{6+(-11)}{2}+(10)\frac{-11+(-4)}{2}\\&\int_{-10}^{20}f(x)\approx-37\end{align*}$

Report an Error

Example Question #1108 : Ap Calculus Ab

$\begin{align*}&\text{Approximate }\int_{4}^{11}f(x)\\&\text{Using the trapezoidal rule, if }f(x)\text{ has the values:}\\&f(4)=-17\\&f(6)=18\\&f(10)=-4\\&f(11)=-19\end{align*}$

Possible Answers:

-9.3333

17.5

Correct answer:

17.5

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(4)=-17\\&f(6)=18\\&f(10)=-4\\&f(11)=-19\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }2,4,1\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{4}^{11}f(x)\approx(2)\frac{-17+(18)}{2}+(4)\frac{18+(-4)}{2}+(1)\frac{-4+(-19)}{2}\\&\int_{4}^{11}f(x)\approx17.5\end{align*}$

Report an Error

Example Question #1109 : Ap Calculus Ab

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[2,22]:\\&f(2)=7\\&f(5)=-15\\&f(7)=-20\\&f(11)=2\\&\text{Approximate the integral of }f(x)\text{ on this interval using the trapezoidal rule.}\end{align*}$

Possible Answers:

-203.3333

-1129.6296

-56.4815

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(2)=7\\&f(5)=-15\\&f(7)=-20\\&f(11)=2\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }3,2,4\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{2}^{22}f(x)\approx(3)\frac{7+(-15)}{2}+(2)\frac{-15+(-20)}{2}+(4)\frac{-20+(2)}{2}\\&\int_{2}^{22}f(x)\approx-83\end{align*}$

Report an Error

Example Question #71 : Numerical Approximations To Definite Integrals

Use the trapezoidal rule to approximate the following integral:

$\int_{0}^{3}(\frac{x^2}{3}+xe^x)dx$

Possible Answers:

$\frac{9-9e^3}{2}$

$\frac{9+9e^3}{2}$

$\frac{3+3e^3}{2}$

${9+9e^3}$

Correct answer:

$\frac{9+9e^3}{2}$

Explanation:

The trapezoidal rule for approximating a definite integral is given by

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(a)+f(b)}{2})$

Using the above formula to approximate our integral, we get

$\int_{0}^{3}(\frac{x^2}{3}+xe^x)dx \approx (3-0)(\frac{3+3e^3}{2})=\frac{9+9e^3}{2}$

Report an Error

Example Question #72 : Numerical Approximations To Definite Integrals

Solve using the trapezoidal approximation:

$\int_{0}^{2}(x^5+e^x)dx$

Possible Answers:

33+e^2

16+e^2

$\frac{33+e^2}{2}$

32+e^2

Correct answer:

33+e^2

Explanation:

The trapezoidal approximation of a definite integral is given by the following:

$\int_{a}^{b}f(x)dx \approx \frac{f(b)+f(a)}{2}(b-a)$

Using this approximation for our integral, we get

$(2-0)(\frac{32+e^2+1}{2})=33+e^2$

Report an Error

Example Question #73 : Numerical Approximations To Definite Integrals

Solve the integral using the trapezoidal approximation:

$\int_{0}^{4}xe^{2x}dx$

Possible Answers:

8e^8

Correct answer:

8e^8

Explanation:

To approximate the definite integral using the trapezoidal rule, we use the following approximation:

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)+f(a)}{2})$

For our integral, we get

$\int_{0}^{4}xe^{2x}dx \approx (4-0)(\frac{4e^8}{2}+0)=8e^8$

Report an Error

Example Question #301 : Integrals

Approximate the integral using the trapezoidal rule:

$\int_{0}^{6}x^2\sin(\frac{\pi x}{12})dx$

Possible Answers:

216

108

Correct answer:

108

Explanation:

To approximate the definite integral using the trapezoidal rule, we use the following approximation:

$\int_{a}^{b}f(x)dx\approx (b-a)(\frac{f(b)+f(a)}{2})$

For our integral, we get

$\int_{0}^{6}x^2\sin(\frac{\pi x}{12})dx \approx \frac{(6)(36+0)}{2}=108$

Report an Error

View AP Calculus AB Tutors

Lawrence
Certified Tutor

Harvey Mudd College, Bachelor of Engineering, Engineering, General. UC Berkeley, Master's/Graduate, Engineering.

View AP Calculus AB Tutors

Timothy
Certified Tutor

Bethune-Cookman University, Bachelor of Science, Mathematics. University of Phoenix, Master of Science, Computer and Informat...

View AP Calculus AB Tutors

Nathan
Certified Tutor

California Polytechnic State University, San Luis Obispo, Bachelor, Electrical Engineering.

All AP Calculus AB Resources

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

Popular Subjects

Reading Tutors in Philadelphia, Physics Tutors in Miami, SSAT Tutors in Denver, Chemistry Tutors in Phoenix, Physics Tutors in Atlanta, ISEE Tutors in San Diego, Math Tutors in San Diego, Biology Tutors in Denver, SSAT Tutors in Seattle, LSAT Tutors in Atlanta

Popular Courses & Classes

ACT Courses & Classes in Phoenix, ACT Courses & Classes in Miami, SAT Courses & Classes in Seattle, MCAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Chicago, SAT Courses & Classes in Philadelphia, SSAT Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Philadelphia, GRE Courses & Classes in Boston, Spanish Courses & Classes in Phoenix

Popular Test Prep

MCAT Test Prep in Houston, SAT Test Prep in San Francisco-Bay Area, SAT Test Prep in Seattle, ACT Test Prep in Phoenix, MCAT Test Prep in Boston, GRE Test Prep in Phoenix, SAT Test Prep in Philadelphia, ACT Test Prep in Chicago, GMAT Test Prep in Washington DC, SSAT 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

AP Calculus AB : Trapezoidal sums

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #31 : Trapezoidal Sums

Example Question #32 : Trapezoidal Sums

Example Question #1106 : Ap Calculus Ab

Example Question #1107 : Ap Calculus Ab

Example Question #1108 : Ap Calculus Ab

Example Question #1109 : Ap Calculus Ab

Example Question #71 : Numerical Approximations To Definite Integrals

Example Question #72 : Numerical Approximations To Definite Integrals

Example Question #73 : Numerical Approximations To Definite Integrals

Example Question #301 : Integrals

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: