Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : 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 #21 : Trapezoidal Sums

$\begin{align*}&\text{Approximate }\int_{2}^{34}f(x)\\&\text{Using the trapezoidal rule, if }f(x)\text{ has the values:}\\&f(2)=-11\\&f(10)=-18\\&f(18)=11\\&f(26)=7\\&f(34)=9\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(2)=-11\\&f(10)=-18\\&f(18)=11\\&f(26)=7\\&f(34)=9\\&\text{The total interval has a length of }32\\&\text{Notice the points are equidistant, with }4\text{ intervals between.}\\&\text{Applying the rules above, we can write:}\\&\int_{2}^{34}f(x)\approx\frac{32}{2\cdot4}[(-11)+2(-18)+2(11)+2(7)+(9)]\\&\int_{2}^{34}f(x)\approx-8\end{align*}$

Report an Error

Example Question #291 : Integrals

$\begin{align*}&f(3)=-7\\&f(13)=13\\&f(23)=12\\&f(33)=14\\&f(43)=6\\&\text{Approximate, using a trapezoidal sum, }\int_{3}^{43}f(x)\end{align*}$

Possible Answers:

3080

770

385

Correct answer:

385

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(3)=-7\\&f(13)=13\\&f(23)=12\\&f(33)=14\\&f(43)=6\\&\text{The total interval has a length of }40\\&\text{Notice the points are equidistant, with }4\text{ intervals between.}\\&\text{Applying the rules above, we can write:}\\&\int_{3}^{43}f(x)\approx\frac{40}{2\cdot4}[(-7)+2(13)+2(12)+2(14)+(6)]\\&\int_{3}^{43}f(x)\approx385\end{align*}$

Report an Error

Example Question #291 : Integrals

$\begin{align*}&f(2)=20\\&f(6)=-19\\&f(12)=1\\&f(16)=-17\\&\text{Approximate, using a trapezoidal sum, }\int_{2}^{16}f(x)\end{align*}$

Possible Answers:

-16.5

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(2)=20\\&f(6)=-19\\&f(12)=1\\&f(16)=-17\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }4,6,4\\&\text{This means we should calculate each trapezoid separately, like in the middle summation term:}\\&\int_{2}^{16}f(x)\approx4\frac{20+(-19)}{2}+6\frac{-19+(1)}{2}+4\frac{1+(-17)}{2}\\&\int_{2}^{16}f(x)\approx-84\end{align*}$

Report an Error

Example Question #67 : Numerical Approximations To Definite Integrals

$\begin{align*}&f(-4)=20\\&f(4)=18\\&f(14)=7\\&f(18)=20\\&\text{Approximate, using a trapezoidal sum, }\int_{-4}^{28}f(x)\end{align*}$

Possible Answers:

331

63.1579

2021.0526

480

Correct answer:

331

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(-4)=20\\&f(4)=18\\&f(14)=7\\&f(18)=20\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }8,10,4\\&\text{This means we should calculate each trapezoid separately, like in the middle summation term:}\\&\int_{-4}^{28}f(x)\approx(8)\frac{20+(18)}{2}+(10)\frac{18+(7)}{2}+(4)\frac{7+(20)}{2}\\&\int_{-4}^{28}f(x)\approx331\end{align*}$

Report an Error

Example Question #1095 : Ap Calculus Ab

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[5,29]:\\&f(5)=-6\\&f(7)=16\\&f(15)=-2\\&f(25)=-4\\&f(29)=-12\\&\text{Approximate the integral of }f(x)\text{ on this interval using the trapezoidal rule.}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(5)=-6\\&f(7)=16\\&f(15)=-2\\&f(25)=-4\\&f(29)=-12\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }2,8,10,4\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{5}^{29}f(x)\approx(2)\frac{-6+(16)}{2}+(8)\frac{16+(-2)}{2}+(10)\frac{-2+(-4)}{2}+(4)\frac{-4+(-12)}{2}\\&\int_{5}^{29}f(x)\approx4\end{align*}$

Report an Error

Example Question #63 : Numerical Approximations To Definite Integrals

$\begin{align*}&\text{Approximate }\int_{2}^{27}f(x)\\&\text{Using the trapezoidal rule, if }f(x)\text{ has the values:}\\&f(2)=-8\\&f(4)=-13\\&f(14)=-7\\&f(16)=-12\\&f(22)=17\end{align*}$

Possible Answers:

-30.692

-171.875

-767.2991

Correct answer:

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(2)=-8\\&f(4)=-13\\&f(14)=-7\\&f(16)=-12\\&f(22)=17\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }2,10,2,6\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{2}^{27}f(x)\approx(2)\frac{-8+(-13)}{2}+(10)\frac{-13+(-7)}{2}+(2)\frac{-7+(-12)}{2}+(6)\frac{-12+(17)}{2}\\&\int_{2}^{27}f(x)\approx-125\end{align*}$

Report an Error

Example Question #21 : Trapezoidal Sums

$\begin{align*}&\text{The function }f(x)\text{ has the following values on the interval }x=[-4,8]:\\&f(-4)=20\\&f(0)=13\\&f(2)=-1\\&f(8)=16\\&\text{Approximate the integral of }f(x)\text{ on this interval using the trapezoidal rule.}\end{align*}$

Possible Answers:

120

123

360

Correct answer:

123

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(-4)=20\\&f(0)=13\\&f(2)=-1\\&f(8)=16\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }4,2,6\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{-4}^{8}f(x)\approx(4)\frac{20+(13)}{2}+(2)\frac{13+(-1)}{2}+(6)\frac{-1+(16)}{2}\\&\int_{-4}^{8}f(x)\approx123\end{align*}$

Report an Error

Example Question #292 : Integrals

$\begin{align*}&f(-9)=9\\&f(1)=6\\&f(9)=14\\&f(13)=-4\\&f(23)=10\\&\text{Approximate, using a trapezoidal sum, }\int_{-9}^{23}f(x)\end{align*}$

Possible Answers:

25.5

205

204

816

Correct answer:

205

Explanation:

$\begin{align*}&\text{A trapezoidal integral approximation over an interval [a, b]}\\&\text{with n subintervals follows the form:}\\&\int_a^b f(x)dx \approx\sum_{i=1}^{n}\frac{b-a}{n}\frac{f(x_{i+1})+f(x_i)}{2}=\frac{b-a}{2n}(f(x_1)+2f(x_2)+2f(x_3)+...+f(x_n))\\&\text{It is essentially a sum of n trapezoids, each with a base of}\\&\text{length :}\frac{b-a}{n}\text{ and variable paired heights :}f(x_i),f(x_{i+1})\text{, which depend on the function values at }x_i,x_{i+1}\\&\text{We're given a set of values:}\\&f(-9)=9\\&f(1)=6\\&f(9)=14\\&f(13)=-4\\&f(23)=10\\&\text{Notice that the points are not equidistant, with varying interval lengths between}\\&\text{These lengths in sequence are: }10,8,4,10\\&\text{This means we should calculate each trapezoid separately: }\sum_{i=1}^{n}(x_{i+1}-x_i)\frac{f(x_{i+1})+f(x_i)}{2}\\&\int_{-9}^{23}f(x)\approx(10)\frac{9+(6)}{2}+(8)\frac{6+(14)}{2}+(4)\frac{14+(-4)}{2}+(10)\frac{-4+(10)}{2}\\&\int_{-9}^{23}f(x)\approx205\end{align*}$

Report an Error

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

5.5

121

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 #31 : 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:

-23.101

-0.74519

-3.875

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

View AP Calculus AB Tutors

Samantha
Certified Tutor

Cornell University, Bachelor of Science, Computer Science.

View AP Calculus AB Tutors

Foysal
Certified Tutor

Towson University, Bachelor, Physics . Virginia Commonwealth University, Master of Science, Health Physics.

View AP Calculus AB Tutors

Bindu
Certified Tutor

All AP Calculus AB Resources

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

Popular Subjects

Chemistry Tutors in Miami, Biology Tutors in Houston, Spanish Tutors in Houston, French Tutors in Dallas Fort Worth, French Tutors in Los Angeles, GRE Tutors in Philadelphia, Biology Tutors in Atlanta, LSAT Tutors in Houston, ISEE Tutors in San Francisco-Bay Area, ACT Tutors in San Diego

Popular Courses & Classes

LSAT Courses & Classes in Denver, Spanish Courses & Classes in New York City, ACT Courses & Classes in Philadelphia, ACT Courses & Classes in Seattle, SAT Courses & Classes in Miami, GRE Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Philadelphia, ACT Courses & Classes in Los Angeles, ISEE Courses & Classes in Atlanta, GRE Courses & Classes in New York City

Popular Test Prep

ACT Test Prep in Los Angeles, SAT Test Prep in Atlanta, MCAT Test Prep in Phoenix, ISEE Test Prep in San Diego, GRE Test Prep in Houston, GMAT Test Prep in Los Angeles, LSAT Test Prep in Philadelphia, MCAT Test Prep in Boston, GRE Test Prep in New York City, MCAT Test Prep in Denver

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 : Integrals

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #21 : Trapezoidal Sums

Example Question #291 : Integrals

Example Question #291 : Integrals

Example Question #67 : Numerical Approximations To Definite Integrals

Example Question #1095 : Ap Calculus Ab

Example Question #63 : Numerical Approximations To Definite Integrals

Example Question #21 : Trapezoidal Sums

Example Question #292 : Integrals

Example Question #31 : Trapezoidal Sums

Example Question #31 : Trapezoidal Sums

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: